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Barnes1  Brief  Histories— The  United  States  History  ;  and  others  to  follow. 
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and  others,  with  Notes,  Lexicons,  Maps,  Illustrations,  Ac. — The  most  complete  and  elegant  editions. 


I 


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Dram 

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National  Educational  Publishers, 
111   &  113  WILLIAM  STREET.  NEW  YORK. 


PRACTICAL 


MATHEMATICS, 


DRAWING   AND   MENSURATION, 


APPLIED    TO 


THE  MECHANIC  ARTS. 


BY  CHARLES  DAVIES,  LL.D., 

AUTHOR    OF 

FiHST    LESSONS     IN    ARITHMETIC;    ARITHMETIC  J     UNIVERSITY    ARITHMETIC 

ELEMENTARY    ALGEBRA;      ELEMENTARY     GEOMETRY;      ELEMENTS 

OF  SURVEYING  ;    ELEMENTS   OF   ANALYTICAL   GEOMETRY  ; 

DESCRIPTIVE   GEOMETRY  ;   SHADES,  SHADOWS,  AND 

LINEAR  PERSPECTIVE  ;  AND  DIFFERENTIAL 

AND    INTEGRAL  CALCULUS. 


A.    S.    BARNES    &   COMPANY, 

NEW  YORK  AND  CHICAGO. 

*f 

1875. 


DAVIES' MATHEMATICS. 

fa®  viit  F@ini  ©@wssa8 

And  Only  Thorough  and  Complete  Mathematical  Series. 


IIST     THREE     PAETS. 


/.    COMMON  SCHOOL    COURSE. 

Davies'  Primary  Arithmetic— The  fundamental  principles  displayed  in 

the  Object  Lessons. 
Davies'  Intellectual  Arithmetic. — Referring  all  operations  to  the  unit  1  as 

the  only  tangible  basis  for  logical  development. 
Davies'  Elements  of  Written  Arithmetic. A  practical   introduction 

to  the  whole  subject.    Theory  subordinated  to  Practice. 
Davies'  Practical  Arithmetic*  —  The  most  successful  combination  of 

Theory  and  Practice,  clear,  exact,  brief,  and  comprehensive. 

//.  ACADEMIC  COURSE. 

Davies'  University  Arithme tic*— Treating  the  subject  exhaustively  as 

a  science,  in  a  logical  series  of  connected  propositions. 
Davies'  Elementary  Alyebra.*— A  connecting  link,  conducting  the  pupil 

easily  from  arithmetical  processes  to  abstract  analysis. 
Davies'  University  Alyebra.*— For  institutions  desiring  a  more  complete 

but  not  the  fullest  course  in  pure  Algebra. 
Davies'  Practical  dlathem  a  tics. —The  science  practically  applied  to  the 

useful  arts,  as  Drawing,  Architecture,  Surveying,  Mechanics,  etc. 
Davies'  Elementary  Geometry. — The  important  principles  in  simple  form, 

but  with  all  the  exactness  of  vigorous  reasoning. 
Davies'  Elements  of  Survey inr/.— Re-written  in  1870.     The  simplest  and 

most  practical  presentation  for  youths  of  12  to  16. 

///.  COLLEGIATE  COURSE. 

Davies'  Bourdon's  Alyebra.* — Embracing  Sturm's  Theorem,  and  a  most 
exhaustive  and  scholarly  course. 

Davies'  University  Alyebra.*— A  shorter  course  than  Bourdon,  for  Insti- 
tutions have  less  time  to  give  the  subject. 

Davies'  Eeyendre's  Geometry.— Acknowledged  the  only  satisfactory  trea- 
tise of  its  grade.    300,000  copies  have  been  sold. 

Davies'  Analytical    Geometry   and    Calculus.— The  shorter  treatises, 
combined  in  one  volume,  are  more  available  for  American  courses  of  study. 

Davies'  Analytical  Geometry.  )The  original  compendiums,  for  those  de- 

Davies'  Diff.  &  Int.  Calculus.  >      siring  to  give  full  time  to  each  branch. 

Davies'  Descriptive  Geometry.— With  application  to  Spherical  Trigonome- 
try, Spherical  Projections,  and  Warped  Surfaces. 

Davies'  Shades,  Shadows,  and  Perspective— A  succinct  exposition  of 
the  mathematical  principles  involved. 

Davies'  Science  of  Mathematics. — For  teachers,  embracing 

I.  Grammar  of  Arithmetic,       TIT.  Logic  and  Utility  of  Mathematics, 
II.  Outlines  of  Mathematics,    IV.  Mathematical  Dictionary. 


*  Keys  may  be  obtained  from  the  Publishers  by  Teachers  only. 

Entered,  according  to  Act  of  Congress,  in  the  year  1852,  by 

CHARLES     DAVIES, 

In  the  Clerk's  Office  of  the  District  Court  of  the  United  States  for  the  Southern  District 

of  New  York. 

FRAC.  MATH. 


PREFACE. 


The  design  of  the  present  work  is  to  afford  an  ele- 
mentary text-book  of  a  practical  character,  adapted  to  the 
wants  of  a  community,  where  every  day  new  demands 
arise  for  the  applications  of  science  to  the  useful  arts 
There  is  little  to  be  done,  in  such  an  undertaking,  ex- 
cept to  collect,  arrange,  and  simplify,  and  to  adapt  the 
work,  in  all  its  parts,  to  the  precise  place  which  it  is 
intended  to  fill. 

The  introduction  into  our  schools,  within  the  last  few 
years,  of  the  subjects  of  Natural  Philosophy,  Astronomy, 
Mineralogy,  Chemistry,  and  Drawing,  has  given  rise  to 
a  higher  grade  of  elementary  studies ,  and  the  extended 
applications  of  the  mechanic  arts  call  for  additional  in- 
formation among  practical  men. 

To  understand  the  most  elementary  treatise  on  Natu- 
ral Philosophy,  or  the  simplest  work  on  the  Mechanic 
Arts,  or  even  to  make  a  plane  drawing,  some  knowledge 
of  the  principles  of  Geometry  is  indispensable ;  and  yet, 
those  in  whose  hands  such  works  are  generally  placed, 
or  who  are  called  upon  to  make  plans  in  the  mechanic 
arts,  feel  that  they  have  hardly  time  to  go  through  with 
a  full  course  of  exact  demonstration. 

The  system  of  Geometry  is  a  connected  chain  of  rig- 
orous logic.  Every  attempt  to  compress  the  reasoning, 
by  abridging  it  at  the  expense  of  accuracy,  has  been  uni- 
formly and  strongly  condemned. 


IV  PREFACE. 

All  the  truths  of  Geometry  necessary  to  carry  out  fully 
the  plan  of  the  present  work,  are  made  accessible  to  the 
general  reader,  without  departing  from  the  exactness  of 
the  geometrical  methods.  This  has  been  done  by  omit- 
ting the  demonstrations  altogether,  and  relying  for  the 
impression  of  each  particular  truth  on  a  pointed  question 
and  an  illustration  by  a  diagram.  In  this  way,  it  is  "be- 
lieved  that  all  the  important  properties  of  the  geometrical 
figures  may  be  learned  in  a  few  weeks  ;  and  after  these 
properties  are  developed  in  their  practical  applications, 
the  mind  receives  a  conviction  of  their  truth  little  short 
of  what  is  afforded  by  rigorous  demonstration. 

The  work  is  divided  into  seven  Books,  and  each  book 
is  subdivided  into  sections. 

In  Book  I.  the  properties  of  the  geometrical  figures 
are  explained  by  questions  and  illustrations. 

In  Book  II.  are  explained  the  construction  and  uses 
of  the  various  scales,  and  also  the  construction  of  geo- 
metrical figures.  It  is,  as  its  title  imports,  Practical 
Geometry. 

Book  III.  treats  of  Drawing. — Section  L,  of  the  Ele 
meats  of  the  Art ;  Section  II.,  of  Topographical  Draw 
ing  ;  and  Section  III.,  of  Plan-Drawing. 

Book  IV.  treats  of  Architecture, — explaining  the  dif- 
ferent orders,  both  by  descriptions  and  drawings. 

Book  V.  contains  the  application  of  the  principles  of 
Geometry  to  the  mensuration  of  surfaces  and  solids.  A 
separate  rule  is  given  for  each  case,  and  the  whole  is 
illustrated  by  numerous  and  appropriate  examples. 

Book  VI.  is  the  application  of  the  preceding  parts  to 
Artificers'  Work.  It  contains  full  explanations  of  all  the 
scales  and  measures  used  by  mechanics — the  construc- 
tion of  these  scales — the  uses  to  which  they  are  applied 


PREFACE.  V 

— and  specific  rules  for  the  calculations  and  computa- 
tions which  are  necessary  in  practical  operations. 

Book  VII.  is  an  introduction  to  Mechanics.  It  ex- 
plains the  nature  and  properties  of  matter,  the  laws  of 
motion  and  equilibrium,  and  the  principles  of  all  the  sim- 
ple machines. 

Book  VIII.  embraces  a  description  of  the  Table  of 
Logarithms  and  their  applications  to  many  practical 
questions ;  also,  many  problems  on  the  measurement  of 
heights  and  distances,  and  an  article  on  the  Strength  of 
Materials,  adopted  from  Chambers'  Educational  Course. 

From  the  above  explanations,  it  will  be  seen  that  the 
work  is  entirely  practical  in  its  objects  and  character 
Many  of  the  examples  have  been  selected  from  a  small 
work  somewhat  similar  in  its  object,  recently  published 
in  Dublin,  by  the  Commissioners  of  National  Education, 
and  some  from  a  small  French  work  of  a  similar  charac 
ter  Others  have  been  taken  from  Bonnycastle's  Men 
suration,  and  the  Library  of  Useful  Knowledge  was 
freely  consulted  in  the  preparation  of  Book  VII.  A 
friend,  Lt.  Richard  Smith,  also  furnished  most  of  the 
first  and  second  sections  of  Book  III. ;  and  the  third 
section  was  chiefly  taken  from  an  English  work. 

The  author  has  indulged  the  hope  that  the  present 
work,  together  with  his  First  Lessons  in  Arithmetic  for 
Beginners,  his  Arithmetic,  Elementary  Algebra,  and  Ele 
mentary  Geometry,  will  form  an  elementary  course  of 
mathematical  instruction  adapted  to  the  wants  of  Prac- 
tical men,  Academies  and  the  higher  grade  of  schools. 

Fishkill  Landing,  ^September,  1852.  * 


DAVIES' 

COURSE  OF  MATHEMATICS. 


DAVIES'  FIRST  LESSONS  IN  ARITHMETIC— For  beginner* 

DAVIES'  ARITHMETIC— Designed  for  the  use  of  Academies  and 

Schoola 

KEY  TO  DAVIES'  ARITHMETIC. 

DAVIES'  UNIVERSITY  ARITHMETIC— Embracing  the  Science 
of  Numbers,  and  their  numerous  applications. 

KEY  TO  DAVIES'  UNIVERSITY  ARITHMETIC. 

DAVIES'  ELEMENTARY  ALGEBRA— Being  an  Introduction  to 
the  Science,  and  forming  a  connecting  link  between  Arithmetic  and 
Algebra. 

KEY  TO  DAVIES'  ELEMENTARY  ALGEBRA. 

DAVIES'  ELEMENTARY  GEOMETRY— This  work  embraces  the 
elementary  principles  of  Geometry.  The  reasoning  is  plain  and  con- 
cise, but  at  the  same  time  strictly  rigorous. 

DAVIES'  PRACTICAL  MATHEMATICS*,  WITH  DRAWING  AND 

MENSURATION— Applied  to  the  Mechanic  Arts. 

DAVIES'  BOURDON'S  ALGEBRA— Including  Sturms'  Theorem,-  - 
Being  an  Abridgment  of  the  work  of  M.  Bourdon,  with  the  addition  of 
practical  examples. 

DAVIES'    LEGENDRE'S   GEOMETRY  and  TRIGONOMETRY 

— Being  an  Abridgment  of  the  work  of  M.  Legendre,  with  the  addition 
of  a  Treatise  on  Mensuration  of  Planes  and  Solids,  and  a  Table  of 
Logarithms  and  Logarithmic  Sines. 

DAVIES'  SURVEYING— With  a  description  and  plates  of  the  Theod- 
olite, Compass,  Plane-Table,  and  Level  :  also,  Maps  of  the  Topo- 
graphical Signs  adopted  by  the  Engineer  Department — an  explana- 
tion of  the  method  of  surveying  the  Public  Lands,  and  an  Elementary 
Treatise  on  Navigation. 

DAVIES'  ANALYTICAL  GEOMETRY— Embracing  the  Equa- 
tions of  the  Point  and  Straight  Line — of  the  Conic  Sections — of 
the  Line  and  Plane  in  Space — also,  the  discussion  of  the  General 
Equation  of  the  second  degree,  and  of  Surfaces  of  the  second  order. 

DAVIES'  DESCRIPTIVE  GEOMETRY,— With  its  application  to 
Spherical  Projections. 

DAVIES'  SHADOWS*  and  LINEAR  PERSPECTIVE. 

DAVIES'  DIFFERENTIAL  and  INTEGRAL  CALCULUS 


CONTENTS. 


BOOK  I.— SECTION  I. 


_•"" 


Page 


Of  Lines  and  Angles 13 

Of  Parallel  Lines — Oblique  Lines 14 

Of  Horizontal  Lines — Vertical  Lines 14 

Of  Angles  formed  by  Straight  Lines — By  Curves 15 

Of  the  Right  Angle — Acute  Angle — Obtuse  Angle 15—16 

Two  Lines  intersecting  each  other  16 — 11 

Parallels  cut  by  a  third  Line — Oblique  Lines 17 

Ol  the  Circle,  and  Measurement  of  Angles 17 

Degrees  in  a  Right  Angle — Quadrant 18 

Sum  of  the  Angles  on  the  same  Side  of  a  Line 19 

Sum  of  the  Angles  about  a  Point 19 

SECTION  II. 

Plane  Figures 20 

Different  Kinds  of  Polygons 21 

Different  Kinds  of  Quadrilaterals 20—22 

Diagonal  of  a  Quadrilateral 23 

Square  on  the  Hypothenuse  of  a  Right-angled  Triangle  23 

SECTION  III. 

Of  the  Circle,  and  Lines  of  the  Circle 24 

Radius  of  the  Circle — Diameter  of  the  Circle 24 

Arc — Chord — Segment — Sector  25 

Angle  at  the  Centre — At  the  Circumference 26 

Angle  in  a  Segment — Secant — Tangent 26 

Figure  inscribed  in  a  Circle — Figure  circumscribed  about  it 27 

Measure  of  an  Angle  at  the  Centre — At  the  Circumference  . . .  27 — 28 

Sum  of  the  Angles  of  a  Triangle— Chords  of  ihc  Circle 28 — 29 


V1I1  CONTENTS. 

BOOK  II.— SECTION  I. 

Tape 

Practical  Geometry 30 

Description  of  Dividers,  and  Uso 30 — 31 

Ruler  and  Triangle,  and  Use 32 — 33 

Scale  of  Equal  Parts,  and  Use 33 — 34 

Diagonal  Scale  of  Equal  Parts,  and  Uso 36—37 

Scale  of  Chords,  and  Use 38 

Protractor,  and  Use 38 

Gunter's  Scale 39 

Practical  Problems.. 40—50 

Questions  for  Practice 50 — 52 

BOOK  III.— SECTION  I. 

Drawing  in  General 53 

Illustration  of  Form— Of  Shade  and  Shadow 53—60 

Manner  of  using  the  Pencil 60 — 61 

General  Remarks  61 — 63 

SECTION  II. 

Topographical  Drawing..... 63 

Description  of  Topographical  Drawing 63 

Explanation  of  the  Figures  and  Signs 64 — 70 

SECTION  III. 

Plan  Drawing 70 

Geometrical  Drawings — Denned 70 

Horizontal  Plane — Denned 70 

Vertical  Plane — Defined 71 

Plan — Denned 71 

Illustrations  of  Plan 71—78 

Sections  78—82 

The  Elevation 82—86 

Remarks  on  Elevations 86* — 88 

Oblique  Elevations 88 — 95 

General  Remarks 95 — 96 

BOOK  IV.— SECTION  I. 

Of  Architecture  97 

Definition  of  Architecture — How  divided 97 

Elements  of  Architecture — Mouldings 97 — 100 


CONTENTS.  IX 

SECTION  IL 

Page 

Orders  of  Architecture — Their  Parts 102 — 104 

Tuscan  Order : 105 

Doric  Order ..  105 

Ionic  Order ,  107 

Corinthian  Order \. ]07 

BOOK  V.— SECTION  I. 

Mensuration  of  Surfaces  109 

Unit  of  Length,  or  Linear  Unit 109 

Unit  of  Surface,  or  Superficial  Unit 109 

Meaning  of  the  term  Rectangle 110 

Denominations  in  which  Areas  are  computed 1 12 — 1 14 

Area  of  the  Triangle ' 114 — 117 

Properties  of  the  Right-angled  Triangle  117 — 119 

Area  of  the  Square 119—120 

Area  of  the  Parallelogram 120 — 121 

Area  of  the  Trapezoid  121—122^ 

Area  of  the  Quadrilateral — Of  an  Irregular  Figure 122 — 1 25 

Areas  and  Properties  of  Polygons 125 — 132 

Of  the  Circle— Area  and  Properties 132—143 

Of  Circular  Rings 143—144 

Area  of  the  Ellipse 144—145 

SECTION  II. 

Mensuration  of  Solids 145 

Definition  of  a  Solid— Different  Kinds 145—147 

Content  of  Solids— Unit  of  Solidity— Table 147—149 

Of  the  Prism  149 — 152 

Of  the  Pyramid 152—157 

Of  the  Frustum  of  a  Pyramid  157 — 159 

Of  the  Cylinder 159—163 

Of  the  Cone 163—166 

Of  the  Frustum  of  a  Cone 167—169 

Of  the  Sphere 169—173 

Of  Spherical  Zones „ 174 

Of  Spherical  Segments 1?4 — 176 

Of  the  Spheroid 176—178 

Of  Cylindrical  Rings 178—179 

Of  tho  Five  Regular  Solids  179—183 

1* 
\ 


X  CONTENTS. 

BOOK  VI. 

Pag» 

Artificers'  Work , 184 

SECTION  I. 

Of  Measures  184 

Carpenters'  Rule — Description  and  Uses 184 — 187 

To  multiply  Numbers  by  the  Carpenters'  Rule 187 — 190 

To  find  the  Content  of  a  Piece  of  Timber  by  the  Carpenters' 

Rule 190 

Table  for  Board  Measure 191 

Board  Measure  192 

SECTION  II. 

Of  Timber  Measure 193 

To  find  the  Area  of  a  Plank 193—194 

To  cut  a  given  Area  from  a  Plank 195 

To  find  the  solid  Content  of  a  square  Piece  of  Timber 195 — 197 

To  cut  off*  a  given  Solidity  from  a  Piece  of  Timber  197 

To  find  the  Solidity  of  round  Timber 198 

Of  Logs  for  Sawing 199 

To  find  the  Number  of  Feet  of  Boards  which  can  be  sawed 

from  a  Log 200 

SECTION  III. 

Bricklayers'  Work 201 

How  Artificers'  Work  is  computed 201 

Dimensions  of  Brick  202 

To  find  the  Number  of  Bricks  necessary  to  build  a  given  Wall  202 

Of  Cisterns 204 

To  find  the  Content  of  a  Cistern  in  Hogsheads 204 

Having  the  Height  of  a  Cistern,  to  find  its  Diameter  that  it 

may  contain  a  given  Quantity  of  Water 205 

Having  the  Diameter,  to  find  the  Height 205 

SECTION  IV. 

Masons' Work  20G 

SECTION  V. 

Carpenters'  and  Joiners'  Work 207 

Of  Bins  for  Grain 203 


CONTENTS.  XI 

Page 

To  fiud  the  Number  of  cubic  Feet  in  any  Number  of  Bushels  208 
To  find  the  Number  of  Bushels  which  a  Bin  of  a  given  Size 

will  hold 208 

To  find  the  Dimensions  of  a  Bin  which  shall  contain  a  given 

Number  of  Bushels.... 209 

SECTION  VI. 

Slaters'  and  Tilers'  Work 210 

SECTION  VII. 

Plasterers'  Work 210 

To  find  the  Area  of  a  Cornice 211 

SECTION  VIII. 

Painters'  Work  212 

SECTION  IX. 

Pavers'  Work 212 

SECTION  X. 

Plumbers' Work  213 

BOOK  VII. 

Introduction  to  Mechanics 215 

SECTION  I. 

Of  Matter  and  Bodies  215 

Matter— Defined 215 

Body— Denned  215 

Space — Defined 215 

Of  the  Properties  of  Bodies 215 

Impenetrability — Defined 215 

Extension — Defined  2l(i 

Figure— Defined . 2lfi 

Divisibility— Defined 216 

Inertia — Defined 217 

Atoms— Defined 217 

Attraction  of  Cohesion  217 

Attraction  of  Gravitation  218 

Weight— Defined  219 


Xll  CONTENTS. 

SECTION  II. 

Page 

Laws  of  Motion,  and  Centre  of  Gravity 219 

Motion — Defined 219 

Force  or  Power — Defined 219 

Velocity— Defined 219 

Momentum — Defined 221 

Action  and  Reaction — Defined 221 

Centre  of  Gravity— Defined 222 

SECTION  III. 

Of  the  Mechanical  Powers 224 

General  Principles..... 224 

Lever— Different  Kinds 224—227 

Pulley 227—229 

Wheel  and  Axle  230 

Inclined  Plane 231 

Wedge— Screw 232 

General  Remarks  233 

SECTION  IV. 

Of  Specific  Gravity 234 

Specific  Gravity — Defined 234 

When  a  Body  is  specifically  heavier  or  lighter  than  another  234 

Density— Defined 234 

To  find  the  Specific  Gravity  of  a  Body  heavier  than  Water  236 

To  find  the  Specific  Gravity  of  a  Body  lighter  than  Water  237 

To  find  the  Specific  Gravity  of  Fluids 238 

Table  of  Specific  Gravities 239 

To  find  the  Solidity  of  a  Body  when  its  Weight  and  Specific 

Gravity  are  known 240 

BOOK  VIII. 

Applications  of  Mathematics 241 

Logarithms 241-  253 

Applications  to  Heights  and  Distances 253- -271 

Strength  of  Materials 271  —296 


GEOMETRY. 


BOOK  I. 

SECTION  I. 

OF    LINES    AND    ANGLES. 

1.  What  is  a  line? 

A  Line  is  length,  without  breadth  or  thickness. 

2.  What  are  the  extremities  of  a  line  called  ? 

The  Extremities  of  a  Line  are   called  points ;    and   any 
place  between  the  extremities,  is  also  called  a  point. 

3.  What  is  a  straight  line  ? 

A  Straight  Line,  is  the  shortest  dis- 
tance from  one  point  to  another.  Thus, 
AB  is  a  straight  line,  and  the  shortest 
distance  from  A  to  B. 

4.  What  is  a  curve  line  ? 

A  Curve  Line,  is  one  which  changes 
its    direction    at    every   point.      Thus,      . 
ABC  is  a  curve  line. 

5.  What  does  the  word  line  mean  ? 

The  word  Line,  when  used  by  itself,  means  a  straight 
line;  and  the  word  Curve,  means  a  curve  line. 


14  BOOK    I. SECTION    I. 

6.  What  is  a  surface  ? 

A  Surface  is  that  which  has  length  and  breadth,  without 
height  or  thickness. 

7.  What  is  a  plane,  or  plane  surface  ? 

A  Plane  is  that  which  lies  even  throughout  its  whole  ex 
tent,  and  with  which  a  straight  line,  laid  in  any  direction, 
will  exactly  coincide. 

8.  When  are  lines  said  to   be  parallel? 

Two  straight  lines  are  said  to  be  paral- 
lel when  they  are  at  the  same  distance      — 

from  each  other  at  every  point.     Parallel 

^nes  will  never  meet  each  other. 

9.  When  are  two  curves  said  to  be  parallel? 
Two  curves  are  said  to  be  parallel  or 

concentric,  when  they  are  at  the  same  dis- 
tance from  each  other.  Parallel  curves 
will  not  meet  each  other. 

10.  What  are  oblique  lines? 

Oblique  lines  are  those  which  ap- 
proach each  other,  and  meet  if  suffi- 
ciently prolonged. 

11.  What  are  horizontal  lines? 

Lines  which  are  parallel  to  the  horizon,  or  to  the  water 
level,  are  called  Horizontal  Lines.  Thus,  the  eaves  of  a 
house  are  horizontal. 

12.  What  are  vertical  lines? 

All  plumb  lines  are  called  Vertical  Lines.  Thus,  trees 
and  plants  grow  in  vertical  lines. 

13.  What  is  an  angle?     How  is  it  read? 

An  Angle  is  the  opening  or  inclination  of  two  lines  which 


OF    LINES    AND    ANGLES. 


^ 


15 


meet  each  other  in  a  point.  Thus  the 
lines  AC,  AB,  form  an  angle  at  the  point 
A.  The  lines  AC,  and  AB,  are  called 
the  sides  of  the  angle,  and  their  intersec- 
tion A,  the  vertex. 

The  angle  is  generally  read  by  placing  the  letter  at  the 
vertex  in  the  middle :  thus,  we  say  the  angle  CAB.  We 
may,  however,  say  simply,  the  angle  A. 


14.  May   angles   be  formed   by  curved 
lines  1 

Yes,   either   by  two    curves,    CA,  BA    A 
forming  the  angle  A,  called  a  curvilinear 
angle : 


Or,  by  two  curves  AC,  AB,  forming 
the  angle  A: 


£ 


Or,  by  a  straight  line  and  curve,  which 
is   called  a  mixtilinear  angle. 


D 


15.   When  is  one  line  said  to  be  perpendicular  to  another  ? 

One  line  is  perpendicular  to  another, 
when  it  inclines  no  more  to  the  one 
side  than  to  the  other.  Thus,  the  line 
DB  is  perpendicular  to  AC,  and  the 
angle  DBA   is  equal  to  DBC. 


A 


~W 


16 


BOOK    I. SECTION    I. 


16.    What  are  the  angles  called? 

When  two  lines  are  perpendicular  to 
each  other,  the  angles  which  they  form 
are  called  right  angles.  Thus,  DBA 
and  DBC  are  right  angles.  Hence,  all 
right  angles  are  equal  to  each  other. 


B 


~fi 


17.  What  is  an  acute  angle? 

An  acute  angle  is  less  than  a  right 
angle.     Thus,  DBC  is  an  acute  angle. 

18.  What  is  an  obtuse  angle? 

An  obtuse  angle  is  greater  than  a 
right  angle.  Thus,  EBC  is  an  obtuse 
angle. 


D 
~C 


B 


19.  If  two  lines  intersect  each  other,  what  follows? 


If  two  lines  intersect  each  other, 
the  opposite  angles  A  and  A  are 
called  vertical  angles.  These  an- 
gles are  equal  to  each  other,  and 
so  also  are  the  opposite  angles  B 
and  B. 


iy4_ 


20.   What  follows  when  two  parallel  lines  are  cut  by  a  third 
line  ? 

If  two  parallel  lines   CD,  AB,  are 


*- 


D 


cut   by  a  third   line   IG,  the   angles 

IHD  and  AFG,  are   called   alternate  > 

angles.      These   angles  are    equal  to   ^      /H 

each  other.     The  angle  IHD  is  also         G 

equal  to  the  angle  IFB,  and  to  the  opposite  angle  CHG. 

21.   What  follows  when  a  line  is  perpendicular  to  one  of 
several  parallel  lines  ? 

If  a   line  be  perpendicular  to  one  of  several  parallel 


OF    LINES    AND    ANGLES. 


17 


E 


H 


lines,  it  will  be  perpendicular  to  all 
the  others.  Thus,  if  AB,  CD,  and 
EF,  be  parallel,  the  line  CH  drawn 
perpendicular  to  AB,  will  also  be  per- 
pendicular to  CD  and  EF. 

22.  How  many  lines  can  be  drawn  from  one  point  pcrpen 
dxcular  to  a  line? 

From  the  same  point  D,  only  one 
line  DB,  can  be  drawn,  which  will 
be  perpendicular  to  AB.  £  2?    q     y 

23.  If  oblique  lines  are  also  drawn,  what  follows  ? 
If  oblique  lines  be  drawn,  as  DC,  DF*  then: — 

1st.  The  perpendicular  DB,  will  be  shorter  than  any  of 
the  oblique  lines. 

2d.  The  oblique  lines  which  are  nearest  the  perpendic- 
ular, will  be  less  than  those  which  are  more  remote. 


OF    THE    CIRCLE    AND    MEASUREMENT    OF    ANGLES. 


24.  What  is  the  circumference  of  a  circle  1 
The   circumference  of  a   circle   is  a 

curve  line,  all  the  points  of  which  are 
equally  distant  from  a  certain  point 
within,  called  the  centre.  Thus,  if  all 
the  points  of  the  curve  AEB  are  equal- 
ly distant  from  the  centre  C,  this  curve 
wrill  be  the  circumference  of  a  circle. 

25.  For  what  is  the  circumference  of  a  circle  used? 
The  circumference  of  a  circle  is  used  for  the  measure- 
ment of  angles. 

26.  How  is  it  divided? 

It  is  divided  into  360  equal  parts  called  degrees,  each 


18 


BOOK    I. SECTION    I. 


degree  is  divided  into  60  equal  parts  called  minutes,  and 
each  minute  into  60  equal  parts  called  seconds.  The  de- 
grees, minutes,  and  seconds,  are  marked  thus,  °,  /,  " ;  and 
9°  18'  10",  are  read,  9  degrees,  18  minutes,  and  10  sec- 
onds. 

27.  How  are  the  angles  measured? 

Suppose  the  circumference  of  a  cir- 
cle to  be  divided  into  360  equal  parts, 
beginning  at  the  point  B.  If,  through 
the  point  of  division  marked  40,  we 
draw  CE,  then,  the  angle  ECB  will 
be  equal  to  40  degrees.  If  we  draw 
CF  through  the  point  of  division  marked  80,  it  will  make 
CB  an  angle  equal  to  80  degrees. 

28.  How  many  degrees  are   there  in  one  right  angle, — in 
two — in  three — in  four  ? 

If  two  lines  AB,  DE,  are  perpen- 
dicular to  each  other,  the  four  angles 
BCD,  DC  A,  ACE,  and  ECB,  will  be 
equal.  These  two  lines  will  divide  the 
circumference  of  the  circle  into  the 
four  equal  parts  BD,  DA,  AE,  and 
EB,  and  each  part  will  measure  one 
of  the  right  angles.  But  one  quarter  of  360  degrees,  is  90 
degrees.  Hence,  one  right  angle  contains  90  degrees,  two 
right  angles  180  degrees,  three  right  angles  270  degrees, 
and  four  right  angles  360  degrees. 

29.  What   is   one   quarter   of  the   circumference   called?--' 
one  half  of  it  ? 

One  quarter  of  the  circumference  is 
called  a  quadrant,  and  contains  90  de-i 
grees.  One  half  of  the  circumference 
is  called  a  semi-circumference,  and  con- 


OF    LINES    AND    ANGLES. 


19 


tains  180  degrees.     Thus,  AC  is  a  quadrant,  and  ACB  is 
a  semi-circumference. 

30.  If  one  straight  line  meets  another,  what  is  the  sum 
of  the  two  angles  equal  to  ? 

If  a  straight  line  EB  meets  anoth- 
er straight  line  AC,  the  sum  of  the 
angles  ABE  and  EBC,  will  be  equal 
to  two  right  angles,  since  these  two 
angles  are  measured  by  half  the  circumference. 

31.  If  there   be   several    angles,    what   will    their    sum   be 
equal  to  ? 

If  there   be    several   angles    CBF,  e  n 

FBE,  EBB,  DBA,  formed  on  the 
same  side  of  a  line,  their  sum,  for  a 
like  reason,  will  be  equal  to  two  right 


32.    What   is   the  sum  of  all   the  angles  formed  about   a 
point  equal  to? 

The  sum  of  all  the  angles  ACB,  BCD, 
DC  A,  which  can  be  formed  about  any 
point  as  C,  is  equal  to  four  right  angles, 
or  360  degrees,  since  they  are  measured 
by  the  entire  circumference. 


20  BOOK    I. SECTION   IX. 


SECTION  II. 

OF    PLANE    FIGURES. 

1.  What  is  a  plane  figure  ? 

A  plane  figure  is  a  portion  of  a  plane,  terminated  on  all 
sides  by  lines,  either  straight  or  curved. 

2.  When  the  bounding  lines  are  straight,  what  is  it  called? 
If  the  bounding  lines  are  straight,  the  space  they  enclose 

is  called  a  rectilineal  figure,  or  polygon. 

3.  What  are  the  lines  themselves  called? 

The  lines  themselves,  taken  together,  are  called  the  pe- 
rimeter of  the  polygon.  Hence,  the  perimeter  of  a  polygon 
is  the  sum  of  all  its  sides. 

4.  Name  the  different  kinds  of  polygons. 


A  polygon  of  three  sides,  is  called  a 
triangle. 


A  polygon  of  four  sides,  is  called  a 
quadrilateral. 


A  polygon  of  five  sides,  is  called  a 
pentagon. 


OF    PLANE    FIGURES. 


21 


A  polygon  of  six   sides,  is  called  a 
hexagon. 


A  polygon  of  seven  sides,  is  called  a  heptagon. 

A  polygon  of  eight  sides,  is  called  an  octagon. 

A  polygon  of  nine  sides,  is  called  a  nonagon. 

A  polygon  of  ten  sides,  is  called  a  decagon. 

A  polygon  of  twelve  sides,  is  called  a  dodecagon. 

5.  What  is  the  perimeter  of  a  polygon  ? 

The  perimeter  of  a  polygon  is  the  sum  of  all  its  sides 

6.  What  is  the  least  number  of  straight  lines  which  can 
enclose  a  space? 

Three  straight  lines,  are  the  smallest  number  which  can 
enclose  a  space. 


7.  Name  the  several  kinds  of  triangles. 

First. — An  equilateral  triangle,  which 
has  its  three  sides  all  equal. 


JD 


Second. — An  isosceles  triangle,  which 
has  two  of  its  sides  equal. 


Third  — A  scalene  triangle,  which  has 
its  three  sides  all  unequal.  I        / 


fi/Wtf  A  ALA 


fs 


22 


BOOK    I. — SECTION    II. 


Fourth. — A  right-angled  triangle,  which 
has  one  right  angle.  In  the  right-angled 
triangle  BAC,  the  side  BC  opposite  the 
right  angle,  is  called  the  hypothenuse. 


B 


8.  What  is  the  base  of  a  triangle  ? — what  its  altitude  ? 
The  base  of  a  triangle   is   the   side  on  which  it  stands 

Thus,  B A  is  the  base  of  the  right-angled  triangle  BAC. 
The  line  drawn  from  the  opposite  angle  perpendicular  to 
the  base,  is  called  the  altitude.     Thus,  A  C  is  the  altitude. 

9.  Name  the  different  kinds  of  quadrilaterals. 


First. — The  square,  which  has  all  its 
sides  equal,  and  all  its  angles  right  an- 
gles. 

Second. — The  rectangle,  which  has  its 
angles  right  angles,  and  its  opposite  sides 
equal  and  parallel. 

Third. — The  parallelogram,  which  has 
its  opposite  sides  equal  and  parallel,  but 
its  angles  not  right  angles. 

Fourth. — The  rhombus,  which  has  all 
its  sides  equal,  and  the  opposite  sides 
parallel,  without  having  its  angles  right 
angles. 

Fifth. — The  trapezoid,  which  has  only 
two  of  its  sides  parallel. 


10.    What  is  the  base  of  a  figure  ?      What  its  altitude  ? 
The  base  of  a  figure  is  the  side  on  which  it  stands,  and 


OF    PLANE    FIGURES. 


23 


the  altitude  is  a  line  drawn  from  the  top,  perpendicular  to 
the  base. 


11.   What  is  a  diagonal? 

A  diagonal,  is  a  line  joining  the 
vertices  of  two  angles  not  adjacent. 
Thus,  AB  is  a  diagonal. 


12.  What  is  the  square  described  on  the  hypothenuse  of  a 
right-angled  triangle  equal  to? 

In  every  right-angled  triangle,  the   square  described  on 
the  hypothenuse,  is  equal  to 
the   sum  of  the  squares  de- 
scribed on  the  other  two  sides. 

Thus,  if  ABC  be  a  right- 
angled  triangle,  right-angled 
at  C,  then  will  the  square  D, 
described  on  AB,  be  equal  to 
the  sum  of  the  squares  E 
and  F,  described  on  the  sides 
CB  and  AC.  This  is  called 
the  carpenter's  theorem. 

By  counting  the  small  squares  in  the  large  square  D, 
you  will  find  their  number  equal  to  that  contained  in  the 
small  squares  F  and  E. 


D 

24 


BOOK    I. SECTION    III. 


SECTION  III. 


OF    THE    CIRCLE,   AND    LINES    OF    THE    CIRCLE. 

1.  What  is  a  circle?     What  is  a  circumference? 
A   circle  is  a  plane   figure,  bounded 

by  a  curve  line,  all  the  points  of  which 
are  equally  distant  from  a  certain  point 
within,  called  the  centre.  The  curve  line 
is  called  the  circumference.  Thus,  the 
space  enclosed  by  the  curve  ABD  is 
called  a  circle :  the  curve  ABD  is  the 
circumference,  and  the  point  C,  the  centre. 

2.  What  is  the  radius  of  a  circle  ?     Are  all  radii  equal  1 
Any  line,  as  CA,  drawn  from  the  cen-  . 

tre  C  to  the  circumference,  is  called  a 
radius,  and  two  or  more  such  Knes,  are 
radii. 

All  the  radii  of  a  circle  are  equal  to 
each  other. 


3.   What  is  the  diameter  of  a  circle  ? 
the  circumference? 

The   diameter  of  a  circle   is  any 
line,   as   AD,   passing   through   the 
centre   and  terminating  in  the 
cumference.     Every  diameter 
circle    divides    it    into    two    equal 
parts,  called  semicircles,  or  half  cir 
cles. 


How  does  it  divide 


OF    THE    CIRCLE,    ETC. 


25 


4.   What  is  an  arc  ? 
An  arc  is  any  portion  of  the  circum- 
ference.    Thus,  AEB  is  an  arc. 


5.   What  is  a  chord? 

A  chord  of  a  circle,  is  a  line  drawn 
within  a  circle,  and  terminating 
circumference,  but  not  passing  through 
the  centre.     Thus,  AB  is  a  chord. 

A  chord  divides  the  circle  into  two 
unequal  parts. 


6.    What  is  a  segment? 

A  segment  of  a  circle,  is  a  part  cut 
off  by  a  chord.  Thus,  AEB  is  a  seg- 
ment.    • 

The  part  AOB,  is  also  a  segment, 
although  the  term  is  generally  applied 
to  the  part  which  is  less  than  a  semi- 
circle. 


7.   What  is  a  sector? 

A  sector  of  a  circle,  is  any  part  of  a 
circle  bounded  by  two  radii  and  the  arc 
included  between  them  Thus,  ACB  is 
a  sector. 

2 


26 


BOOK    I. SECTION    III. 


8.    What  is  an  angle  at  the  centre? 

An  angle  at  the  centre,  is  one  whose 
vertex  is  at  the  centre  of  the  circle. 
Thus,  BCE,  or  ECD,  is  aa  angle  at 
the  centre. 


9.  What  is  an  angle  at  the  circumfe- 
rence ? 

An  angle  at  the  circumference,  is  one 
whose  angular  point  is  in  the  circum- 
ference. Thus,  BAC,  or  BOC,  is  an 
angle  at  the  circumference. 


10.  What  is  an  angle  in  a  seg- 
ment ? 

An  angle  in  a  segment^  is  formed  by 
two  lines  drawn  from  any  point  of  the 
segment  to  the  two  extremities  of  the 
arc.  Thus,  ABE  is  an  angle  in  a  seg- 
ment. 

11.  What  is  a  secant  line? 

A  secant  line,  is  one  which  meets  the 
circumference  in  two  points,  and  lies 
partly  within  and  partly  without.  Thus, 
AB  is  a  secant  line. 


12.    What  is  a  tangent  line? — What  position  has  it  with 
the  radius  passing  through  the  point  of  contact  ? 

A  tangent  is   a  line  which  has  but  one  point  in  com- 


OF    THE    CIRCLE,    ETC 


27 


mon  with  the  circumference.  Thus, 
EMD  is  a  tangent.  The  point  M 
at  which  the  tangent  touches  the 
circumference  is  called  the  point  of 
contact.  The  tangent  line  is  perpen- 
dicular to  the  radius  passing  through 
the  point  of  contact.  Thus,  CM  is 
perpendicular  to  EMD. 


13.   When  is  a  figure  said  to  be  inscribed  in  a  circle? — 
What  is  said  of  the  circle  ? 

A  figure  is  said  to  be  inscribed  in 
a  circle  when  all  the  angular  points 
of  the  figure  are  in  the  circumference. 
The  circle  is  then  said  to  circumscribe 
the  figure.  Thus,  the  triangle  ABC  is 
inscribed  in  the  circle,  and  the  circle 
circumscribes  the  triangle. 


14.    When  is  a  figure  said   to   be  circumscribed  about   a 
circle  ? 

A  figure  is  said  to  be  circumscribed 
about  a  circle,  when  all  the  sides  of  the 
figure  touch  the  circumference.  The  cir- 
cle is  then  said  to  be  inscribed  in  the 
figure. 


15.  How  is  an  angle  at  the  centre  of  a  circle  measured? 

An  angle  at  the  centre  of  a  circle  is 
measured  by  the  arc  contained  by  the 
sides  of  the  angle.  This  arc  is  said  to 
subtend  the  angle.  Thus,  the  angle  ACB 
is  measured  by  the  degrees  in  the  arc 
AEBi  and  is  subtended  by  the  arc  AEB. 


2b 


BOOK    I. SECTION   III. 


16.    What  measures  an  angle  at  the  circumference? 

An  angle  at  the  circumference  of  a  circle,  is  measured 
by  half  the  arc  which  subtends  it. 
Thus,  the  angle  BAD  is  measured  by 
half  the  arc  BD.  Hence,  it  follows, 
that  when  an  angle  at  the  centre  and 
an  angle  at  the  circumference  stand 
on  the  same  arc  BD,  the  angle  at  the 
centre  will  be  double  the  angle  at  the 
circumference. 


]  7.  What  is  the  sum  of  the  three  an- 
gles of  any  triangle  equal  to  ? 

To  180  degrees,  since  they  will  be 
measured  by  one  half  of  the  entire  cir- 
cumference. 


18.    What  is  an  angle  in  a  semicircle  equal  to? 

An  angle  inscribed  in  a  semicircle,  is  a  right  angle. 
Thus,  if  AB  be  the  diameter  of  a  cir- 
cle, then  will  the  angle  ACB  be  equal 
to  90  degrees.  This  angle  is  measured 
by  one  half  the  semi-circumference,  that 
is,  by  one  half  of  180°,  or  by  90°. . 


19.  Are   the   arcs  intercepted  by  parallel  chords  equal,  or 
unequal  ? 

Two  parallel  chords  intercept  equal 
arcs.  That  is,  if  the  chords  AB  and 
CD  are  parallel,  the  arcs  AC  and 
DB,  which  they  intercept,  will  be 
equal. 


OF    THE    CIRCLE,    ETC. 


29 


20.  If  cu  line  be  drawn  from  the  centre  of  a  circle  perpen- 
dicular to  a  chord,  what  follows  ? 

If  from  the  centre  of  a  circle 
a  line  be  drawn  perpendicular  to 
a  chord,  it  will  bisect  the  chord, 
and  also  the  arc  of  the  chord. 
Thus,  CFE  drawn  from  the  cen- 
tre C,  perpendicular  to  AB,  bi- 
sects AB  at  F,  and  also  makes 
AE  =  EB. 


21.  How  is  the  distance  from  the  centre  of  a  circle  to  a 
chord  measured? 

The  distance  from  the  centre  of  a  circle  to  a  chord,  is 
measured  on  a  perpendicular  to  the  chord. 


22.   How  are   chords  which   are  equally  distant  from  the 
centre  ? 

In  the  same,  or  in  equal  circles, 
chords  which  are  equally  distant  from 
the  centre,  are  equal.  Thus,  if  CA 
=  CB,  then  will  the  chord  FG  = 
chord  DE.  \l 


30 


BOOK    II. SECTION    I. 


BOOK  II. 


SECTION   I. 


PRACTICAL    GEOMETRY. 


1.  What  is  Practical  Geometry? 

Practical  geometry  explains  the  methods  of  constructing, 
or  describing  the  geometrical  figures. 

2.  What  is  a  problem  ? 

Any  question  which  requires  something  to  be  done ;  and 
doing  the  thing  required,  is  called  the  solution  of  the  prob- 
lem. 

3.  What  are  necessary  in  the  solution  of  geometrical  prob- 
lems ? 

Certain  instruments  which  are  now  to  be  described. 

4.  What  are  the  dividers  or  compasses  ? 


The  dividers  is  the  most  simple  and  useful  of  the  in- 


PROBLEMS  31 

struments  used  for  describing  figures.  It  consists  of  two 
legs,  ba  and  be,  which  may  be  easily  turned  arourd  a 
joint  at  b. 

5.  How  will  you  lay  off  on  a  line,  as  CD,  a  distance  equal 
to  AB? 

Take  up  the  dividers  with  the  thumb  and  second  finger, 
and  place  the  fore-finger  on  the  joint  at  b.     Then,  set  one 
foot  of  the  dividers  at  A,  and  ex- 
tend the  other  leg  with  the  thumb       t -? 


and  fingers,  until  the  foot  reaches       c  U      n 

to  B.     Then,  raising  the  dividers, 

place  one  foot  at  C,  and  mark  with  the  other  the  distance 
CE,  this  will  evidently  be  equal  to  AB. 

6.  How  will  you  describe  from  a  given  centre,  the  circum- 
ference of  a  circle  having  a  given  radius  ? 

Let  C  be  the  given  centre,  and  CB  the  given  radius. 
Place  one  foot  of  the  dividers  at  C,  and 
extend  the  other  leg  until  it  shall  reach 
to  B.  Then  turn  the  dividers  around  the 
leg  at  C,  and  the  other  leg  will  describe 
the  required  circumference. 

7.  How  may  this*  be  done  on  a  black  board  with  a  string 
and  chalk  ? 

Take  one  end  of  the  string  between  the  thumb  and  fore- 
finger of  the  left  hand,  and  place  it  at  the  centre  C.  Then 
take  the  length  of  the  radius  on  the  string,  at  which  point 
place  the  chalk  held  between  the  thumb  and  finger  of  the 
right  hand.  Then,  holding  the  end  of  the  string  firmly  at 
C,  turn  the  right  hand  around,  and  the  chalk  will  trace 
the  circumference  of  the  circle. 


32  BOOK    II. SECTION    I. 

8-  Describe  the  ruler  and  triangle. 


A  ruler  of  a  convenient  size,  is  about  twenty  inches  in 
length,  two  inches  wide,  and  one-fifth  of  an  inch  in  thick- 
ness. It  should  be  made  of  a  hard  material,  perfectly- 
straight  and  smooth. 

The  hypothenuse  of  the  right-angled  triangle,  which  is 
used  in  connection  with  it,  should  be  about  ten  inches  in 
length,  and  it  is  most  convenient  to  have  one  of  the  sides 
considerably  longer  than  the  other.  We  can  resolve  with 
the  ruler  and  triangle  the  two  following  problems. 

9.  Describe  the  manner  of  drawing  through  a  given  point 
a  line,  which  shall  be  parallel  to  a  given  line,  with  the  ruler 
and  triangle. 

Let  C  be  the  given  point,  and  AB  the  given  line. 

Place  the  hypothenuse  of  the  tri-  c 

angle  against  the  edge  of  the  ruler,       l 

and  then  place  the  ruler  and  trian-     ji_ ]$ 

gle  on  the  paper,  so  that  one  of  the 

sides  of  the  triangle   shall  coincide  exactly  with  AB — the 

triangle  being  below  the  line  AB. 

Then  placing  the  thumb  and  fingers  of  the  left  hand 
firmly  on  the  ruler,  slide  the  triangle  with  the  other  hand 
alonor  the  ruler  until  the  side  which  coincided  with  AB 
reaches  the  point  C.     Leaving  the  thumb  of  the  left  hand 


SCALE    OF    EQUAL    PARTS.  33 

on  the  iuler.  extend  the  fingers  upon  the  triangle  and  hold 
it  firmly,  and  with  the  right  hand  mark  with  a  pen  or  pen- 
cil a  line  through  C :  this  line  will  be  parallel  to  AB. 

10.  Explain  the  manner  of  drawing  through  a  given  point, 
a  line  which  shall  be  perpendicular  to  a  given  line,  with  the 
ruler  and  triangle. 

Let  AB  be  the  given  line,  and  D  the  given  point. 

Place,  as  before,  the  hypothenuse 
of  the  triangle  against  the  edge  of  the 
ruler.  Then  place  the  ruler  and  tri- 
angle so  that  one  of  the  sides  of  the      A 

triangle,  shall  coincide  exactly  with 

the  line  AB.  Then  slide  the  triangle  along  the  ruler  until 
the  other  side  reaches  the  point  D.  Draw  through  D  a 
straight  line,  and  it  will  be  perpendicular  to  AB. 

11.  What  is  a  scale  of  equal  parts  ? 

. , I    .7   .s  ..T.A.5  .6  .7  .8  .'J  I? 


A  scale  of  equal  parts  is  formed  by  dividing  a  line  of  a 
given  length,  into  equal  portions. 

If,  for  example,  the  line  ab,  of  a  given  length,  say  one 
inch;  be  divided  into  any  number  of  equal  parts,  as  10,  the 
scale  thus  formed  is  called  a  scale  often  parts  to  the  inch. 

12.    What  is  the  unit  of  a  scale,  and  how  is  it  laid  off? 

The  line  ab  which  is  divided,  is  called  the  unit  of  the 
scale.  This  unit  is  laid  off  several  times  on  the  left  of  the 
divided  line,  and  the  points  marked  1,  2,  3,  &c.  The  unit 
of  scales  of  equal  parts,  is,  in  general,  either  an  inch  oi 
an  exact  part  of  an  inch.  If,  for  example,  the  unit  of  the 
scale  ab,  were  one  inch,  the  scale  would  be  one  of  ten 
parts  to  the  inch ;  if  it  were  half  an  inch,  the  scale  would 

2* 


34  BOOK    II. SECTION    I. 

be  one  of  ten  parts  to  half  an  inch,  or  of  20  parts  to  the 
inch. 

13  How  will  you  take  from  the  scale  two  inches  and  six- 
tenths  ? 

Place  one  foot  of  the  dividers  at  2  on  the  left,  and  ex- 
tend the  other  to  .6,  which  marks  the  sixth  of  the  small 
divisions :  the  dividers  will  then  embrace  the  required  dis- 
tance. 

14.  How  will  you  lay  down,  on  paper,  a  line  of  a  given 
length,  so  that  any  number  of  its  parts  shall  correspond  to  the 
unit  of  the  scale  ? 

Suppose  that  the  given  line  were  75  feet  in  length,  and 
it  were  required  to  draw  it  on  paper,  on  a  scale  of  25  feet 
to  the  inch. 

The  length  of  the  line,  75  feet,  being  divided  by  25,  will 
give  3,  the  number  of  inches  which  will  represent  the  line 
on  paper. 

Therefore,  draw  the  indefinite  line  ABf  on  which  lay  off 

a ' ■ tr° 

a  distance  AC  equal  to  3  inches:  AC  will  then  represent 
the  given  line  of  75  feet,  drawn  to  the  required  scale. 

15.  What  does  the  last  question  explain? 

The  last  question  explains  the  method  of  laying  down  a 
line  upon  paper,  in  such  a  manner  that  a  given  number  of 
parts  shall  correspond  to  the  unit  of  the  scale,  whether  that 
unit  be  an  inch  or  any  part  of  an  inch. 

When  the  length  of  the  line  to  be  laid  down  is  given, 
and  it  has  been  determined  how  many  parts  of  it  are  to 
be  represented  on  the  paper  by  a  distance  equal  to  the 
unit  of  the  scale,  we  find  the  length  which  is  to  be  taken 
from  the  scale  by  the  following 


PROBLEMS.  35 

RULE. 

Divide  the  length  of  the  line  by  the  number  of  parts  which  is 
to  be  represented  by  the  unit  of  the  scale :  the  quotient  will 
show  the  number  of  parts  which  is  to  be  taken  from  the 
scale. 

EXAMPLES 

1.  If  a  line  of  640  feet  in  length  is  to  be  laid  down  on 
paper,  on  a  scale  of  40  feet  to  the  inch ;  what  length  must 
be  taken  from  the  scale  ? 

40)640(16  inches 

2.  If  a  line  of  357  feet  is  to  be  laid  down  on  a  scale 
of  68  feet  to  the  unit  of  the  scale,  (which  we  will  suppose 
half  an  inch,)  how  many  parts  are  to  be  taken  ? 

(  5.25,  parts,  or 
Ans'    (  2.625  inches. 

16.  When  the  length  of  a  line  is  given  on  the  paper,  how 
will  you  find  the  true  length  of  the  line  represented  ? 

Take  the  line  in  the  dividers  and  apply  it  to  the  scale, 
and  note  the  number  of  units,  and  parts  of  a  unit  to  which 
it  is  equal.  Then,  multiply  this  number  by  the  number  of 
parts  which  the  unit  of  the  scale  represents,  and  the  pro- 
duct will  be  the  length  of  the  line. 

EXAMPLES. 

1.  Suppose  the  length  of  a  line  drawn  on  the  paper,  to 
be  3.55  inches,  the  scale  being  40  feet  to  the  inch:  then, 

3.55  X  40  =  142  feet,  the  length  of  the  line. 

2.  If  the  length  of  a  line  on  the  paper  is  6.25  inches, 
and  the  scale  be  one  of  30  feet  to  the  inch,  what  is  the 
true  length  of  the  line  1 

Ans.  187.5  feet. 


36 


BOOK    II. — SECTION    I. 


17.  How  do  you  construct  the  diagonal  scale  of  equal 

PARTS  ? 


rlf 

<? 

• 

dii 

1  /I  1  1  i  1  \  i 

r 

66 

i    w 

07 

/      M  1   1/      / 

Oti 

Ml      Mill 

I 

Oo 

1      n    1 1    1 

04 

1    1  II  II  1 

03 

1  f  1/  /  1  /  /  / 1 

oi 

i  1  r- 

1 

ul 

17   f-l 

/  i  M  i 

i 

5 

I 

t 

.2  .2  .3. J 

,5  .6  .7.S.S    A 

This  scale  is  thus  constructed.  Take  ab  for  the  unit 
of  the  scale,  which  may  be  one  inch,  \,  J,  or  £  of  an 
inch,  in  length.  On  ab  describe  the  square  abed.  Divide 
the  sides  ab  and  dc  each  into  ten  equal  parts.  Draw  of 
and  the  other  nine  parallels  as  in  the  figure. 

Produce  ab  to  the  left,  and  lay  off  the  unit  of  the  scale 
any  convenient  number  of  times,  and  mark  the  points  1, 
2,  3,  &c.  Then,  divide  the  line  ad  into  ten  equal  parts, 
and  through  the  points  of  division  draw  parallels  to  ab  as 
in  the  figure. 

Now,  the  small  divisions  of  the  line  ab  are  each  one- 
tenth  (.1)  of  ab ;  they  are  therefore  .1  of  ad,  or  .1  of  ag 
or  gh. 

If  we  consider  the  triangle  adf,  the  base  df  is  one-tenth 
of  ad,  the  unit  of  the  scale.  Since  the  distance  from  a 
to  the  first  horizontal  line  above  ab,  is  one-tenth  of  the 
distance  ad,  it  follows  that  the  distance  measured  on  that 
line  between  ad  and  of  is  one-tenth  of  df:  but  since  one- 
tenth  of  a  'tenth  is  a  hundredth,  it  follows  that  this  distance 
is  one  hundredth  (.01)  of  the  unit  of  the  scale.  A  like  dis- 
tance measured  on  the  second  line  will  be  two  hundredths 
(.02)  of  the  unit  of  the  scale  ;  on  the  third,  .03  ;  on  the 
fourth,  .04,  &c, 


SCALES    OF    EQUAL    PARTS.  37 

18.  Hov)  will  you  take  in  the  dividers  the  unit  of  the  scale 
and  any  number  of  tenths? 

Place  one  foot  of  the  dividers  at  1,  and  extend  the  other 
to  that  figure  between  a  and  b  which  designates  the  tenths. 
If  two   or  more  units  are  required,  the    dividers    must  be 
x^aced  on  a  point  of  division  farther  to  the  left. 

19.  How  do  you  take  off  units,  tenths,  and  hundredths  ? 
Place  one  foot  of  the  ctMders  where  the  vertical  line; 

through  the  point  which  designates  the  units,  intersects  the 
line  which  designates  the  hundredths ;  then,  extend  the 
dividers  to  that  line  between  ad  and  be  which  designates 
the  tenths :  the  distance  so  determined  will  be  the  one 
required. 

For  example,  to  take  off  the  distance  2.34,  we  place  one 
foot  of  the  dividers  at  I,  and  extend  the  other  to  e:  and  to 
take  off  the  distance  2.58,  we  place  one  foot  of  the  dividers 
at  p,  and  extend  the  other  to  q. 

Remark  I. — If  a  line  is  so  long  that  the  whole  of  it  can- 
not be  taken  from  the  scale,  it  must  be  divided,  and  the 
parts  of  it  taken  from  the  scale  in  succession. 

Remark  II. — If  a  line  be  given  upon  the  paper,  its  length 
can  be  found  by  taking  it  ,in  the  dividers  and  applying  it  to 
the  scale. 

20.  How  do  you  construct  a  scale  of  chords  1 

If  with  any  radius,  as  AC,  we  describe  the  quadrant 
AD,  and  then  divide  it  into  90  equal  parts,  each  part  is 
called  a  degree. 

Through  A,  and  each  point  of  division,  let  a  chord  be 
drawn,  and  let  the  lengths  of  these  chords  be  accurately 
laid  ofl  on  a  scale  :  such  a  scale  is  called  a  scale  of  chords. 
In  the  figure,  the  chords  are  drawn  for  every  ten  degrees. 

The  scale  of  chords  being  once   constructed,  the  radius 


38 


BOOK    II.  — SECTION    I. 


of  the  circle  from  which  the  chords  were  obtained,  is  known ; 
for,  the  chord  marked  60  is  always  equal  to  the  radius  of 


the  circle.  A  scale  of  chords  is  generally  laid  down  on 
Gunter's  scale,  and  on  the  scales  which  belong  to  cases  of 
mathematical  instruments,  and  is  marked  cho. 

21.  How  will  you  lay  off  an  angle  with  a  scale  of  chords ; 
say  an  angle  of  30  degrees  ? 

Let  AB  be  the  line  from  which  the  angle  is  to  be  laid 
off,  and  A  the  angular  point. 

Take  from  the  scale,  the  chord  of  60 
degrees,  and  with  this  radius  and  the 
point  i  as  a  centre,  describe  the  arc 
BC.  Then  take  from  the  scale  the 
chord  of  the  given  angle,  say  30  de- 
grees, and  with  this  line  as  a  radius,  and  B  as  a  centre, 
describe  an  arc  cutting  BC  in  C.  Through  A  and  C  draw 
the  line  AC,  and  BAC  will  be  the  required  angle. 

22.  Describe  the  semicircular  protractor. 

This  instrument  is  used  to  lay  down,  or  protract  angles. 
It  may  also  be  used  to  measure  angles  included  between 
lines  already  drawn  upon  paper. 

It  consists  of  a  brass  semicircle  ACB,  divided  to  half 


SEMICIRCULAR  PROTRACTOR. 


39 


degrees.     The  degrees  are  numbered  from  0  to  180,  both 
ways;  that  is,  from  A  to  B,  and  from  B  to  A.     The  di- 


visions, in  the  figure,  are  only  made  to  degrees.  There  is 
a  small  notch  at  the  middle  of  the  diameter  AB,  which  in- 
dicates the  centre  of  the  protractor. 

23-  How  do  you  lay  off  an  angle  with  a  protractor? 

Place  the  diameter  AB  on  the  line,  so  that  the  centre 
shall  fall  on  the  angular  point.  Then  count  the  degrees 
contained  in  the  given  angle  from  A  towards  B,  or  from  B 
towards  A,  and  mark  the  extremity  of  the  arc  with  a  pin. 
Remove  the  protractor,  and  draw  a  line  through  the  point 
so  marked  and  the  angular  point :  this  line  will  make  with 
the  given  line  the  required  angle. 

24.  Describe  gunter's  scale. 

This  is  a  scale  of  two  feet  in  length,  on  the  faces  of 
which,  a  variety  of  scales  are  marked.  The  face  on  which 
the  divisions  of  inches  are  made,  contains,  however,  all  the 
scales  necessary  for  laying  down  lines  and  angles.  These 
are,  the  Scale  of  Equal  Parts,  the  Diagonal  Scale  of  Equal 


40 


BOOK    II. SECTION 


Parts,  and  the   Scale  of  Chords,  all  of  which  have  been 
described. 

25.  How  do  you  bisect  a  given  straight  line  ;  that  is,  divide 
it  into  two  equal  parts  1 

Let  AB  be  the  given  line.  With  A 
as  a  centre,  and  a  radius  greater  than 
half  of  AB,  describe  an  arc  IFE.  Then 
remove  the  foot  of  the  dividers .  from  A 
to  B,  and  with  the  same  radius  describe 
the  arc  EHI.  Then  join  the  points  I 
and  E  by  the  line  IE:  the  point  D, 
where  it  intersects  AB,  will  be  the  mid- 
dle of  the  line  AB. 


/; 

/ 
/ 

/ 

\ 

4     H\ 

U,F 

B 

26.  At  a  given  point  in  a  given  straight  line,  how  do  you 
draw  a  perpendicular  to  that  line  ? 

Let  A  be  the  given  point,  and  BC  the  given  line. 

From  A  lay  off  any  two  distances 
AB  and  AC,  equal  to  each  other. 
Then,  from  the  points  B  and  C,  as 
centres,  with  a  radius  greater  than 
BA,  describe  two  arcs  intersecting  Jr- 
each  other  in  D:  draw  AD,  and  it 
will  be  the  perpendicular  required. 


D 

t 


SECOND    METHOD. 

27.  When  the  point  A  is  near  the  end  of  the  line 
Place  one  foot  of  the  dividers  at 
any  point,  as  P,  and  extend  the  other 
leg  to  A.  Then  with  P  as  a  centre, 
and  radius  from  P  to  A,  describe  the 
circumference  of  a  circle.  Through 
C,  where  the  circumference  cuts  BA, 
and  the  centre  P,  draw  the  line  CPD. 


PROBLEMS. 


41 


Then  draw  AD,  and  it  will  be  perpendicular  to  CA,  since 
CAD  is  an  angle  in  a  semicircle. 

28.  Draw  from  a  given  point  without  a  straight  line,  a  per- 
pendicular to  that  line. 

Let  A  be  the  given  point,  and  BD 
the  given  line. 

From  the  point  A  as  a  centre,  with 
a  radius  sufficiently  great,  describe  an 
arc  cutting  the  line  BD  in  the  two 
points  B  and  D:  then  mark  the  point 
E,  equally  distant  from  the  points  B  and  D,  and  draw  AE 
and  AE  will  be  the  perpendicular  required. 


SECOND    METHOD. 

29.  When  the  given  point  A,  is  nearly  opposite   one   end 
of  the  given  line. 

Draw  AC  to  any  point,  as  C  of 
the  line  BD.  Bisect  AC  at  F.  Then 
with  Fas  a  centre,  and  FC  or  FA, 
as  a  radius,  describe  the  semicircle 
CD  A.  Then  draw  DA,  and  it  will 
be  perpendicular  to  BD  at  D. 

30.  At  a  point,  in  a  given  line,  to  make  an  angle  equal  to 
a  given  angle. 

Let  A  be  the  given  point,  AE  j  jy 

the  given  line,  and  IKL  the  given 
angle. 

From  the  vertex  K,  as  a  cen- 
tre with  any  radius,  describe  the  arc  IL,  terminating  in 
the  two  sides  of  the  angle.  From  the  point  A  as  a  centre,. 
with  a  distance  AE  equal  to  KI,  describe  the  arc  ED, 
then  take  the  chord  LI,  with  which,  from  the  point  E  as 


42 


BOOK    II. SECTION    I. 


a  centre,  describe  an  arc  cutting  the  indefinite  arc  DE,  in 
D:  draw  AD,  and  the  angle  EAD  will  be  equal  to  the 
given  angle  K. 

31.  How  do  you  divide  a  given  angle,  or  a  given  arc,  into 
two  equal  parts  ? 

Let  C  be  the  given  angle,  and  AEB 
the  arc  which  measures  it. 

From  the  points  A  and  B  as  centres, 
describe  with  the  same  radius  two  arcs 
cutting  each  other  in  D:  through  D 
and  the  centre  C  draw  CD :  the  angle 
ACE  will  be  equal  to  the  angle  ECB,  and  the  arc  AE  to 
the  arc  EB. 


J\ 


D 


m 


b£ 


32.  How  do  you  draw  through  a  given  point  a  line  pareU 
lei  to  a  given  line  ? 

Let  A  be  the  given  point,  and         p  r 

BC  the  given  line. 

From  A  as  a  centre,  with  a  ra- 
dius greater  than  the  shortest  dis- 
tance from  A  to  BC,  describe  the  indefinite  arc  ED:  from 
the  point  E  as  a  centre,  with  the  same  radius,  describe  the 
arc  A F ;  make  ED  =  AF,  and  draw  AD :  then  will  AD 
be  the  parallel  required. 

33.  If  two  angles  of  a  triangle  are  given,  how  do  you  find 
the  third? 

Draw  the  indefinite  line 
DEF.  At  the  point  E,  make 
the  angle  DEC  equal  to  one 
of  the  given  angles,  and  then 
the  angle  CEH  equal  to  the 

other:  the  remaining  angle  HEF  will  be  the  third  angle 
required . 


PROBLEMS. 


43 


34.  If  two  sides  and  the  included  angle  of  a  triangle  are 
given,  how  do  you  describe  the  triangle  1 

Let  the  line  B  =  150  feet,  and       „ 
C  =  120  feet,  be  the  given  sides  ; 
and   A  =  30   degrees,   the    given 
angle :  to  describe  the  triangle  on 
a  scale  of  200  feet  to  the  inch. 

Draw   the   indefinite   line    DG, 
and  at  the  point  D,  make  the  angle 

GDH  equal  to  30  degrees  ;  then  lay  off  DG  equal  to  three 
quarters  of  an  inch,  and  it  will  represent  the  side  B  =  150 
feet :  make  DH  equal  to  six-tenths  of  an  inch,  and  it  will 
represent  C  =  120  feet:  then  draw  GH,  and  GDH  will  be 
the  required  triangle. 

35.  If  the  three  sides  of  a  triangle  are  given,  how  do  you 
describe  the  triangle  1 

Let  A,  B,  and  C,  be  the  sides. 
Draw  DE  equal  to  the  side  JL. 
From  the  point  D  as  a  centre, 
with  a  radius  equal  to  the  sec- 
ond side  B,  describe  an  arc : 
from  E  as  a  centre,  with  a  radius 
equal  to  the  third  side  C,  de- 
scribe another  arc  intersecting  the  former  in  F ;  draw  DF 
and  EF,  and  DEF  will  be  the  triangle  required. 

36.  If  two  sides  of  a  triangle  and  an  angle  opposite  one 
of  them  are  given,  how  do  you  describe  the  triangle  ? 

Let  A  and  B  be  the  given 
sides,  and  C  the  given  angle, 
which  we  will  suppose  is  oppo- 
site the  side  B.  Draw  the  in- 
definite line  DF,  and  make  the 
angle  FDE  equal  to  the  angle 


44  BOOK    II. SECTION    1. 

C :  take  DE  =  A,  and  from  the  point  E  as  a  centre,  with 
a  radius  equal  to  the  other  given  side  B,  describe  an  arc 
cutting  DF  in  F ;  draw  EF :  then  will  DEF  be  the  required 
triangle. 

If  the  angle  C  is  acute,  and        .      

the  side  B  less  than  A,  then      „ ^^^C 

the    arc    described   from   the  „ 

centre    E    with    the     radius  ^v-^^^V 

EF  —  B  will  cut  the  side  DF        ^^Cs  ^y 

in  two  points,  F  and  G,  lying  ^*\.  /& 

on  the  same  side  of  D  :  hence  * ' 

there  will  be  two    triangles, 

DEF  and  DEG,  either  of  which  will  satisfy  all  the  con- 
ditions of  the  problem. 

37.  If  the  adjacent  sides  of  a  parallelogram,  with  the  angle 
which  they  contain,  are  given,  how  do  you  describe  the  paral- 
lelogram ? 

Let  A   and  B  be    the  given  . 

sides,  and  C  the  given  angle.  /  Tr* 

Draw  the  line  DE  =  A ;    at  /  L, 

the    point   D,   make    the    angle      ~? 

EDF  =  C ;  take  DF  =  B :  de-     B)  { 

scribe  two  arcs,  the  one  from  F 

as  a  centre,  with  a  radius  FG  —  DE,  the  other  from  E,  as 

a  centre,  with  a  radius  EG  —  DF ;  through  the  point  G, 

where  these  arcs  intersect  each  other,  draw  FG,  EG ;  then 

DEGF  will  be  the  parallelogram  required. 

38.  How  do  you  describe  the  circumference  of  a  circle  which 
shall  pass  through  three  given  points  ? 

Let  A,  B,  and  C,  be  the  three  given  pomts. 
Join  these  points  by  straight  lines  AB,  BC,  CA.     Then 
bisect  any  two  of  these  straight  lines  by  the  perpendicu- 


PROBLEMS. 


45 


lars  OF,  OD,  as  in  Section  25, 
and  the  point  O,  where  these 
perpendiculars  intersect  each 
other,  will  be  the  centre  of  the 
circle. 

Place  one  foot  of  the  dividers 
at  this  centre,  and  extend  the 
other  to  A,  B,  or  C,  and  then  with 
(his  radius,  let  the  circumference 
be  described. 


39.  How  do  you  find  the  centre  of  a  circle  when  the  circum- 
ference is  given  ? 

Take  any  three  points,  as  A,  B,  and  C,  (see  last  figure,) 
and  join  them  by  the  lines  AB  and  BC.  Then  bisect  these 
lines  by  the  perpendiculars  OD  and  OF,  an 1  O  will  be  the 
centre  of  the  circle. 


40.  How  do  you  divide  a  given  line  AB,  into  any  number 
of  equal  parts  1 

Let  AB  be  the  given  line  to  be 
divided.  Let  it  be  required,  if  you 
please,  to  divide  it  into  five  equal 
parts. 

Throughout  A,  one  extremity  of 
the  line,  draw  A  h,  making  an  angle 

with  AB.  Then  lay  off  on  Ah,  five  equal  parts,  Ac,  cd,  df 
fg,  gh,  after  which  join  h  and  B.  Through  the  points  of 
division  c,  d,  f  and  g,  draw  lines  parallel  to  hB,  and  they 
will  divide  AB  into  the  required  number  of  equal  parts. 

41.  How  do  you  describe  a  square  on  a  given  line  ? 

Let  AB  be  the  given  line.     At  the  point  B,  draw  BC 
perpendicular  to  AB,  and  then  make  it  equal  to  BA. 


46 


BOOK    II. — SECTION    I. 


Then,  with  A  as  a  centre,  and 
radius  equal  to  AB,  describe  an 
arc ;  and  with  C  as  a  centre,  and 
the  same  distance  AB,  describe 
another  arc,  and  through  D,  their 
point  of  intersection,  draw  AD  and 
CD;  then  will  AB  CD  be  the  re- 
quired square. 


DL 


41.  How   do   you  construct  a  rhombus,   having  given   the 
length  of  one  of  the  equal  sides  and  one  of  the  angles  1 

Let  AB  be  equal  to  the  given 
side,  and  E,  the  given  triangle. 

At  B,  lay  off  an  angle  ABC, 
equal  to  E,  and  make  BC  equal 
to  AB.  Then  with  A  and  C  as 
centres,  and  a  radius  equal  to 
AB,  describe  two  arcs,  and 
through  D,  their  point  of  inter- 
section, draw  the  lines  AD  and  CD,  and  ABCD  will  be 
the  required  rhombus. 


42.  How  do  you  inscribe  a  circle  in  a  given  triangle? 

Let  ABC  be  the    given 
triangle. 

Bisect  either  two  of  the 
angles,  as  A  and  C,  by  the 
lines  AO  and  CO,  and  the 
point  of  intersection  O  will 
be  the  centre  of  the  in- 
scribed circle.  Then,  through  the  point  of  intersection  O 
draw  a  line  perpendicular  to  either  side,  and  it  will  be  the 
radius. 


PROBLEMS. 


47 


43.  How  do  you  inscribe  an  equilateral  triangle  in  a  cir- 


^J2 


cle?   • 

With  any  point  i,  as  a  centre,  and 
radius  equal  to  the  radius  of  the  circle, 
describe  an  arc  cutting  the  circum- 
ference in  B  and  C.  Then  bisect  the 
arc  BDC,  after  which,  draw  BC,  BD, 
and  CD,  and  BDC  will  be  an  equi- 
lateral triangle. 

44.  How  do  you  inscribe  a  hexagon  in  a  circle  ? 
Describe   the   equilateral  triangle 

as  before.  Then  bisect  the  arc  CD 
in  jF,  and  the  arc  BD  at  G,  and  draw 
AC,  CF,  FD,  DG,  GB,  and  BA, 
and  ACFDGB  will  be  the  hexagon 
required.  Or  the  hexagon  may  be 
inscribed  by  applying  the  radius  six 
times  around  the  circumference. 

45.  How  do  you  inscribe  a  dodecagon  in  a  circle  ? 

Bisect  the  arcs  which  subtend  the  chords  of  the  hexa- 
gon, and  through  the  points  of  bisection  draw  chords,  and 
there  will  be  formed  a  regular  dodecagon. 

46.  How  do  you  inscribe  in  a  circle  a  regular  pentagon  ? 
Draw  the  diameters  AP  and  MN 

at  right  angles  to  each  other,  and 
bisect  the  radius  ON  at  E.  From 
£asa  centre,  and  EA  as  a  radius, 
describe  the  arc  As ;  and  from  the 
point  iasa  centre,  and  radius  As, 
describe  the  arc  sB. 

Join  the  points  A  and  B,  and  the 
line  AB,  being  applied  five  times 
around  the  circle,  will  form  the  required  pentagon. 


48 


BOOK    II. SECTION 


47.  How  do  you  inscribe  in  a  circle  a  regular  decagon? 
For  the  decagon,  bisect  the  arcs  which  subtend  the  siues 

of  the  pentagon,  and  join  the  points  of  bisection ;  and  the 
lines  so  drawn  will  form  the  regular  decagon. 

48.  How  will  you  inscribe  in  a  circle  a  polygon  having  any 
number  of  sides  ? 

Divide  the  circumference  of  the  circle  into  as  many  equal 
parts  as  there  are  sides  of  the  polygon,  and  draw  lines 
through  the  points  of  division :  these  lines  will  be  the  sides 
of  the  required  polygon. 

49.  How  do  you  inscribe  a  square 
in  a  given  circle  ? 

Let  ABCD  be  the  given  circle. 
Draw  two  diameters  DB  and  AC 
at  right  angles  to  each  other,  and 
through  the  points  A,  B,  C,  and 
D,  draw  the  lines  AB,  BC,  CD, 
and  DA:  then  ABCD  will  be  an 
inscribed  square. 

50.  How  would  you  inscribe  an  octagon? 

By  bisecting  the  arcs  AB,  BC,  CD,  and  DA,  and  join- 
ing the  points  of  bisection,  we  can  form  an  octagon ;  and 
by  bisecting  the  arcs  which  subtend  the  sides  of  the  octa- 
gon, we  can  inscribe  a  polygon  of  sixteen  sides. 

51.  How  will  you  circumscribe  a  square  about  a  circle  ? 
Draw  two  diameters  AB  and  CD 

at  right  angles  to  each  other;  and 
through  their  extremities  A,  B,  C,  and 
D,  draw  lines  respectively  parallel  to 
the  diameters  CD  and  AB :  a  square 
will  thus  be  formed  circumscribing 
the  circle. 


D 


PROBLEMS. 


49 


52.  How  do  you  draw  a  line  which  shall  be  tangent  to  the 
circumference  of  a  circle  at  a  given 
point  ? 

Let  A  be  the  given  point. 
Through.  A  draw  the  radius  AC, 
and  then  draw  DA  perpendicular 
to  the  radius  at  the  extremity  A. 
The  line  DA  will  be  tangent  to 
the  circumference  at  the  point  A. 


53.  How   do  you  draw  through  a  given  point  without  a 
circle  a  line  which  shall  be  tangent  to  the  circumference  ? 

Let  A  be  the  given  point  without 
the  given  circle  BED.  Join  the 
centre  C  and  the  given  point  A, 
and  bisect  the  line  CA  at  O. 

With  O  as  a  centre,  and  OA  as 
a  radius,  describe  the  circumfer- 
ence AB  CD.  Through  B  and  D 
draw  the  lines  AB  and  AD,  and 
they  will  be  tangent  to  the  circle 
BED  at  the  points  B  and  D. 


54.    What  is  an  ellipse? 

It  is  an  oval  curve  ACBD. 


55.  What  is  the  longest  line  which  can  be  drawn  within 
the  curve  called?      What  is  the  shortest  line  called? 

The  longest  line  AB  is  called  the  transverse  axis;  and 
the  shortest  line  DC  is  called  the  conjugate  axis.  The 
point  E,  at  which  they  intersect,  is  called  the  centre  of  the 
ellipse. 

3 


50 


BOOK    II. SECTION    1. 


56.  What  are  the  foci  of 
an  ellipse  ? 

They  are  two  points  F 
and  77,  determined  by  de- 
scribing the  arc  of  a  circle 
with  D  as  a  centre,  and 
a  radius  DF  equal  to  AE, 
half  of  the  transverse  axis. 

57.  How  will  you  describe  an  ellipse  when  you  know  the 
two  axes  AB  and  CD  ? 

First,  find  the  foci  F  and  77  by  describing  an  arc  with 
D  as  a   centre,  and  with  a  radius  equal  to  AE. 

Secondly,  take  a  string  or  thread  equal  in  length  to  AB, 
and  fasten  the  extremities  at  the  foci  F  and  H.  Then 
place  a  pencil  against  the  string  and  move  it  round,  bear- 
ing it  tight  against  the  string,  and  the  point  will  describe 
the  ellipse  ADBC. 


QUESTIONS  TO  BE  PUT  FROM  FIGURES  MADE  BY  THE  TEACHER 
UPON  THE   BLACK-BOARD. 


SECTION 


What  is  a  line  ? 

What  is  a  right  line? 

What  is  a  curve  ? 

What  does  the  word  line  imply  ? 

What  is  a  surface  ? 

What  is  a  plane  ? 

What  are  parallel  right  lines? 

What  are  parallel  curves? 

What  are  oblique  lines? 

What  are  horizontal  lines? 

What  are  vertical  lines? 

What  is  an  angle  ?  how  read  ? 

What  are  curvilinear  angles? 

When  is  one  line  perpendicular  to 

another  ? 
What  are  the  angles  then  called  ? 


What  is  an  acute  angle  ? 

What  is  an  obtuse  angle? 

What  follows  when  two  lines  inter- 
sect each  other? 

What  follows  when  one  line  cuts 
two  parallels? 

What  follows  when  one  line  is  per- 
pendicular to  one  of  several  paral- 
lels? 

How  many  lines  can  be  drawn  from 
a  point  perpendicular  to  a  given 
line? 

If  oblique  lines  are  drawn,  how  do 
they  compare  ? 

What  is  the  circumference  of  a  cir- 
cle? 


QUESTIONS. 


51 


For  what, is  it  used? 

How  is  it  divided  ? 

How  are  angles  measured? 

How  many  degrees  in  one  right  an- 
gle? 

What  is  one  quarter  of  the  circum- 
ference called  ?    One  half? 


When  one  straight  line  meets  an- 
other, what  is  the  sum  of  the 
angles  on  the  same  side? 

If  there  are  several  angles,  what  is 
their  sum  equal  to  ? 

What  is  the  sum  of  all  the  angles 
about  a  given  point  equal  to? 


SECTION    II. 


What  is  a  plane  figure  ? 

What  is  it  called  when  the  bound- 
ing lines  are  straight  ? 

What  are  the  lines  themselves 
called  ? 

What  is  a  triangle  ? 

What  is  a  quadrilateral  ? 

What  is  a  polygon  of  five  sides? 

What  is  a  polygon  of  six  sides  ? 

What  is  a  polygon  of  seven  sides? 

What  is  a  polygon  of  eight  sides  ? 

Wha£  is  a  polygon  of  nine  sides  ? 

What  is  a  polygon  of  ten  sides  ? 

What  is  a  polygon  of  twelve  sides  ? 


What  is  the  smallest  number  of 
straight  lines  which  can  enclose 
a  space  ? 

What  are  the  several  kinds  of  tri- 


What  is  the  base  of  a  triangle  ? 

What  its  altitude? 

What  are  the  different  kinds  of 
quadrilaterals? 

What  is  the  base  of  a  figure  ? 

What  is  a  diagonal  ? 

What  is  the  square  described  on  the 
hypothenuse  of  a  right-angled  tri- 
angle equal  to  ? 


SECTION    III. 


What  is  a  circle  ? 

What  is  a  circumference  » 

What  is  the  radius  of  a  circle 

What  is  an  arc  ? 

What  is  a  chord  ? 

What  is  a  segment? 

What  is  a  sector? 

What  is  an  angle  at  the  centre  ? 

What  is  an  angle  at  the  circum- 
ference ? 

What  is  an  angle  in  a  segment  ? 

What  is  a  secant  line  ? 

What  is  a  tangent  line  ? 

What  position  has  the  tangent  with 
the  radius  ? 


When  is  a  figure  said  to  be  inscribed 

in  a  circle  ? 
When  circumscribed  about  it  ? 
How  is  an  angle  at  the  centre  of  a 

circle  measured  ? 
What  measures  an  angle  at  the  cir- 
cumference ? 
What  is  the  sum  of  the  three  angles 

of  a  triangle  equal  to  ? 
How  does  a  perpendicular  through 

the  centre  divide  the  chord? 
How  do  the  distances  from  the  con- 

tre  to  equal  chords  compare  witli 

each  other? 


52 


BOOK  II.  — SECTION  I. 


PRACTICAL  GEOMETRY. 


What  is  Practical  Geometry  ? 

What  is  a  problem  ? 

What  are  the  dividers? 

How  do  you  lay  off  a  line  ? 

How  do  you  describe  the  circum- 
ference of  a  circle  ? 

How  on  the  black-board  ? 

Describe  the  ruler  and  triangle,  and 
the  manner  of  using  them. 

How  do  you  draw  a  perpendicular  ? 

What  is  a  Scale  of  Equal  Parts? 

What  is  a  unit  of  the  scale  ? 

Explain  how  you  take  from  the 
scale  a  given  number  of  parts. 

Explain  the  Diagonal  Scale. 

What  is  a  Scale  of  Chords? 

How  will  you  lay  off  an  angle? 

What  is  the  Semicircular  Protrac- 
tor? 

How  do  you  lay  off  an  angle  with  it  ? 

Describe  Gunter's  Scale. 

How  do  you  bisect  a  line  ? 

How  do  you  draw  a  perpendicular 
at  a  given  point  ? 

How  do  you  make  an  angle  equal 

'   to  a  given  angle  ? 

How  do  you  bisect  an  arc  ? 

How  do  you  draw  a  parallel  to  a 
given  line  ? 

When  two  angles  of  a  triangle  are 
given,  how  do  you  find  the  third  ? 


When  two  sides  and  the  included 
angle  are  given,  how  do  you  de- 
scribe the  triangle? 

How  do  you  describe  a  parallelo- 
gram with  the  same  given  ? 

How  do  you  pass  the  circumference 
of  a  circle  through  three  points  ? 

How  do  you  divide  a  line  into  any 
number  of  equal  parts  ? 

How  do  you  describe  a  square  ? 

How  do  you  construct  a  rhombus  ? 

How  do  you  inscribe  a  circle  in  a 
given  triangle  ? 

How  do  you  inscribe  an  equilateral 
triangle  in  a  given  circle? 

How  do  you  inscribe  a  hexagon  in 
a  circle  ? 

How  do  you  inscribe  a  dodeca- 
gon? 

How  do  you  inscribe  in  a  circle  a 
polygon  having  any  number  of 
sides  ? 

How  do  you  inscribe  a  square  ?  an 
octagon  ? 

How  do  you  circumscribe  a  square 
about  a  circle  ? 

How  do  you  draw  a  line  tangent  to 
a  circle  at  a  point  of  the  circum- 
ference ? 

How  from  a  point  without  the  cir- 
cumference ? 


Note. — After  the  teacher  shall  havo  made  the  above  figures,  or  most 
of  them,  on  the  black-board,  and  the  pupils  copied  them  on  their  slates, 
let  the  students  then  be  called  to  the  black-board  in  turn,  and  practised 
in  the  drawing  of  them. 


OF   DRAWING    IN    GENERAL.  53 


BOOK  III. 

SECTION   I. 

OF    DRAWING    IN    GENERAL. 

1.  What  are  drawings? 

Drawings  are  representations  to  the  eye  of  the  forms, 
dimensions,  positions,  and  appearance  of  objects.  They 
form  a  written  language,  which  is  easily  comprehended  by 
every  one. 

2.  What  are  the  uses  of  drawing  ? 

Drawing,  to  the  practical  man,  furnishes  a  simple  means 
of  describing  and  explaining  a  thing  in  a  brief  and  striking 
manner.  On  this  account,  alone,  its  great  advantages  are 
everywhere  apparent.  Drawings,  also,  impress  the  mind 
with  images  approaching  nearer  to  the  reality,  than  any 
other  means  of  description.  The  pen  of  the  ablest  historian 
presents  but  a  feeble  image,  when  compared  with  the  pic- 
tured canvass  of  the  painter,  or  the  life-like  forms  of  the 
sculptor. 

3.  When  you  look  at  a  single  object,  what  do  you  observe 
that  distinguishes  it  from  other  objects  ? 

When  we  observe  a  single  object,  we  discover  that  we 
are  able  to  recognise  it  by  means  of  three  properties  which 
distinguish  it  from  other  objects,  viz. :  its  form,  its  light 
and  shade,  and  its  color.     If  we   consider  more   than  one 


54 


BOOK    III. SECTION    I. 


object,  we   are   then  able  to  distinguish  them  from  each 
other,  by  their  relative  position,  also. 

4.  How  do  you  illustrate  the  idea  of  form  1 
If  we  join  any  three  points 

A,  J5,  and  C,  by  straight 
lines,  the  result  will  be  &  figure 
or  form  of  a  triangle.  If  we 
take  another  point  D,  and  join 
the  three  points  A,  D,  and  C, 
we  shall  have  the  form  of  an- 
other triangle  ADC.  The 
straight  lines  which  bound 
each  of  these  figures,  make 
up  what  is  called  its  out- 
line. 

If  with  C  as  a  centre,  and  any  radius,  we  describe  the 
circumference  of  a  circle,  the  curve 
so  drawn  will  be  the  outline  of  the 
circle. 

Now,  the  triangle  and  circle  differ 
from  each  other  only  in  form,  and 
the  form  is  determined  by  the  out- 
line :  hence  we  see  that  outline  is 
one  means  of  representing  form  to 
the  eye.  It  is  thus  that  we  are 
able  to  distinguish  a  triangle  from  a  circle,  and  a  circle 
from  a  square ;  and  the  drawings  of  their  outlines  present 
to  the  mind,  through  the  eye,  the  idea  of  the  objects  them- 
selves. 

5.  How  do  you  illustrate  light  and  shade  ? 

If  we  hold  any  object  in  the  sun's  rays,  it  is  evident,  that 
that  part  of  it  which  is  turned  towards  the  sun  will  be 
lighted ;  and  that  the  part  which  is  turned  away  from  the 


OF    DRAWING      IN    GENERAL. 


55 


sun  will  be  comparatively  dark.     The  part  towards  the  sun 
is   called  the  light;  the  other  part,  the  shade 


6.  In  what  manner  do  light  and  shade  modify  the  idea 
a  form  which  is  represented  only  by  its  outline  ? 

The  circle  whose  centre  is  C, 
is  the  outline  of  so  much  of  the 
flat  white-  paper  as  is  contained 
within  its  circumference. 

Now,  if  we  observe  a  sphere,  or 
perfectly  round  ball,  we  find  that, 
in  every  position,  its  outline  is  also 
a  circle.  We  cannot  tell,  there- 
fore, whether  this  circle  is  the  out- 
line of  a  circular  piece  of  paper  or  of  a  sphere. 


of 


Let  the  circle  whose  centre  is 
D,  be  the  outline  of  a  sphere.  If 
we  suppose  the  light  to  proceed 
from  the  left  hand,  then  the  part 
of  the  sphere  towards  the  left  will 
be  the  light,  and  the  part  towards 
the  right,  the  shade. 


Leaving  the  white  paper  for  the 
light,  we  will  represent  the  shade, 
or  dark  part,  by  means  of  lines 
drawn  in  such  a  manner,  as  to 
darken  that  part  occupied  by  the 
shade. 


56 


BOOK    III. SECTION    I. 


In  a  similar  manner, 
the  outline  of  a  rectangle 
may  be  distinguished  from 
that  of  a  cylinder  by  means 
of  light  and  shade.  Thus 
we  see  that  light  and  shade 
furnish  a  distinction  be- 
tween objects  whose  outlines  are  the  same. 


Cylinder 


7-  In  how  many  ways  may  the  shade  on  a  body  be  modi' 
fied? 

In  two  ways :  viz  ,  in  its  depth  or  intensity,  and  its 
color. 

8.  How  do  you  know  which  part  of  a  body  has  the  greatest 
depth  or  intensity  of  shade  ? 

If  there  were  no  atmosphere,  and  no  body  in  existence 
except  the  one  we  are  considering,  that  part  of  it  which 
does  not  receive  the  sun's  rays  would  be  invisible.  But 
since  the  atmosphere,  as  well  as  every  other  substance  in 
nature,  reflects  back  the  light  which  it  receives,  casting  it 
in  a  direction  contrary  to  that  of  the  sun's  rays ;  it  follows, 
that  the  part  of  any  object  which  does  not  receive  the  direct 
light  of  the  sun,  will  yet  receive  light  from  other  objects, 
behind  it  with  reference  to  the  sun,  and  will  be  sufficiently 
illuminated  to  exhibit  its  form.  Now,  since  bodies  are  more 
or  less  illuminated  as  they  receive  the  light  directly  or  ob- 
liquely, it  follows,  that  if  we  conceive  the  reflecting  body 
to  be  placed  directly  behind  the  one  receiving  the  light,  that 
the  part  nearest  the  reflecting  body  will  receive  more  light 
than  the  parts  more  remote  ;  and  hence,  the  shade  there 
will  be  less  intense.  It  therefore  follows,  that  the  effect 
of  reflected  light  on  the  depth  of  shade,  will  be  the  greatest 
near  the  outline  of  the  body  which  is  farthest  from  the 
source  of  light. 


OF    DRAWING    IN    GENERAL. 


57 


9.  How  may  the  shade  of  an  object  be  modified  in  regard 
to  color  1 

Every  reflected  ray  of  light  is  of  the  same  color  as  the 
body  which  reflects  it,  and  when  such  rays  illuminate  a  dark 
object,  they  also  impart  to  it  their  color.  This  may  be 
shown  by  holding  any  dark  body,  as  a  sphere,  in  the  sun's 
rays,  and  placing  near  it,  and  opposite  to  the  sun,  a  piece  of 
bright-colored  red  or  yellow  paper.  The  reflected  rays  from 
the  paper  will  impart  their  tint  to  the  shade  of  the  sphere. 

10.  What  is  the  shadow  on  a  body  1 

The  shadow  on  a  body  is  that  part  of  it  from  which  the 
light  is  intercepted  by  some  opaque  body. 

11.  How  may  the  forms  of  objects  be 
discovered  by  means  of  the  shadows  which 
they  cast  or  receive  ? 

It  is  evident  that  the  shadow  of  a  tri- 
angle, or  of  a  square,  on  a  flat  surface, 
will,  in  certain  positions,  exactly  resemble 
the  bodies  which  cast  them.  But  the  sur- 
face which  receives  the  shadow  will  modi- 
fy the  shape  of  it ;  and  thus  the  shadow 
will  also  give  an  idea  of  the  form  of  the 
surface  on  which  it  falls. 

For  example,  the  rectangle 
in  the  figure  casts  a  shadow 
of  such  a  shape  on  the  wall 
and  step  which  are  behind  it, 
as   to    show    their    form   dis- 
tinctly.    Without  the  shadow, 
the  two  lines  which  are  the    r 
outlines   of    the    step,    might    - 
equally  well  represent  two  ho-    • 
rizontal  lines  drawn  upon  the  wall. 
3* 


58 


BOOK    III. SECTION    I. 


This  example  exhibits  the 
outline,  light  and  shade,  reflec- 
tion, and  shadow  of  a  cup ; 
and  is  an  illustration  of  the 
foregoing  principles. 

12.  How  may  the  relative  position  of  objects  be  determined 
by  the  shadows  which  they  cast  or  receive? 

When  a  shadow  is  entirely- 
separated  from  the  body  which 
casts  it,  as  is  the  shadow  of  the 
sphere  in  this  example,  it  is  then 
plain  that  a  space  intervenes  be- 
tween the  body  and  the  surface  jm  ■  Wsm^- 
on  which  the  shadow  falls. 

But  when  the  shadow  joins  the  body  which  casts  it,  as 
in  this  example,  then  the   body 
casting  the  shadow  touches  the 
surface    on    which   the    shadow 
falls. 

Hence,  the  nearer  an  object  is 
to  the  surface  on  which  the  shad- 
ow falls,  the  nearer  will  the  shadow  approach  to  the  object. 


OF    DRAWING    IN    GENERAL. 


59 


The  example  of  the  house  shows,  by  the  shadows  on  B 
and  C,  that  B  stands  further  back  than  A,  and  C  farther 
than  B. 

The  shadows  in  the  example  which  follows,  exhibit  the 
difference  between  the  forms  of  three  objects  whose  out- 
lines are  exactly  the  same.  The  shade  on  them  cannot  be 
represented  in  these  outlines. 


ill I!l 


^>l^l 


X 


I 


!!» 


■'■<!! 


13.  What  may  be  said  of  color,  as  a  means  of  distinguish- 
ing objects  from  each  other  ? 

Of  this,  it  is  only  necessary  to  observe,  that  when  we 
have  represented  the  form  of  an  object,  its  light  and  shade, 
and  its  shadow,  if  we  wish  still  further  to  distinguish  it  from 
other  objects,  we  have  but  to  add  its  appropriate  color.  Foi 
example,  in  the  drawing  of  a  machine,  if  we  wish  to  ex- 
hibit the  difference  between  the  wood,  the  iron,  and  the 
brass,  the  natural  colors  of  these  should  be  added  in  the 
drawing. 

14.  What  effect  have  shade  and  shadow  ? 

Shade  and  shadow  have  the  effect  of  obscuring  the  out- 
line, form,  and  color,  of  that  part  of  every  object  on  which 
they  are  found.  Hence  shading,  in  drawing,  is  the  ob- 
scuring, in  imitation  of  nature,  of  those  portions  of  the 
objects  we  are  representing,  and  from  which  the  light  is 
intercepted.  There  is  this  difference,  however,  between 
nature   and  art : — in  the  former  wo  distinguish  and  deter- 


60 


BOOK    III. — SECTION    I. 


mine  forms  by  means  of  the   light ;    in  the  latter,  by  the 
shade  and  shadow. 

15.  By  what  is  the  process  of  shading  regulated? 

The  process  of  shading  a  drawing  varies  according  to 
the  instrument  used.  The  pen  is  capable  of  making  only 
lines  and  dots ;  hence,  if  we  employ  it  only,  we  are  con- 
fined to  those  two  methods  of  shading.  The  brush  and 
lead  pencil  possess,  in  addition  to  the  resources  of  the  pen, 
the  capability  of  laying  a  smooth,  graduated  tint  of  shade, 
which  by  the  brush  may  also  be  made  of  any  color  that 
may  be  desired. 


16.    What  may  be  said  of  the  use  of  the  pencil? 

The  acquisition  of  a  skilful  and  easy  manner  of  hand- 
ling the  pencil,  depends  in  a  great  measure  upon  the  way 
of  holding  it.  The  thumb,  with  the  first  and  second  fingers, 
should  grasp  the  pencil 
about  an  inch  from  its 
point.  The  thumb  should 
not  be  drawn  back,  as 
we  are  taught  in  holding 
a  pen  for  writing ;  but 
should  be  placed  opposite 
to,  or  a  little  below  the  points  of  the  fingers. 

This  position  will  enable  the  hand 
to  move  from  left  to  right,  and  to 
draw  curved  lines  with  as  much  free- 
dom in  that  direction,  as  from  right  to 
left. 

Let  the  learner  now  practise  the 
drawing  of  such  lines  as  are  shown 
in  the  figure,  from  left  to  right. 

In  drawing  straight  lines  by  the  hand,  the  learner  should 


OF    DRAWING    IN    GENERAL. 


61 


■fc^. 


?l«*. 


not  begin  b)&  timidly  drawing 
dotted  lines,  as  is  usually- 
done  ;  but  the  pencil  should 
be  passed  rapidly  two  or  three 
times  from  one  extremity  of 
the  line  to  the  other,  without 
touching  the  paper,  and  then 
the  line  should  be  drawn  at 
one  stroke.  Should  it  not  be 
correct,  repeat  the  trial  until 
it  is  right;  after  which,  and 
not  before,  efface  whatever 
is  wrong. 

In  the  same  manner, 
curved  lines  "may  be  first 
sketched  out  by  drawing 
broken  lines,  and  after- 
wards rounding  off  the  angles  and  effacing  the  straight  lines 
These  distinctions  may  appear  trifling,  and  too  minute,  but 
nothing  is  more  certain  than  that  a  careful  and  intelligent 
observance  of  them,  will  ensure  a  rapid  and  easy  manner 
of  sketching. 


GENERAL    REMARKS. 

It  is  not  intended,  nor  would  it  be  possible,  to  give  here 
more  than  a  few  practical  hints  concerning  the  general  prin- 
ciples of  the  art  of  drawing.  The  learner,  after  familiar- 
izing himself  with  them,  and  with  the  short  directions  as 
to  the  mechanical  part,  should  copy  some  good  drawings, 
under  the  direction  of  an  instructor.  He  should  then  take 
some  simple  object,  such  as  a  book,  a  cup,  an  inkstand, 
<fec,  and  placing  it  before  him,  endeavor  to  describe  its  po- 
sition and  proportions  by  means  of  its  outline.  This  is  done 
by  comparing  the  lines  which  make  up  its  outline  with  each 


62 


BOOK    III. SECTION    I. 


other,  regarding  both  their  comparative  length  and  the  an- 
gles which  they  make  with  each  other.  If  the  direction 
and  relative  length  of  each  line  are  right,  the  drawing  must 
be  correct. 

An  easy  help  in  finding 
the  direction  of  a  line  nearly 
vertical,  is  to  hold  at  arm's 
length,  between  the  eye  and 
object,  (a  pyramid,  for  ex- 
ample,) a  ruler  which  serves 
as  a  plumb  line.  The  edge 
of  the  ruler  being  vertical, 
when  brought  in  range  with 
the  point  A,  will  show  how 
much  the  line  AC  varies 
from  a  perpendicular. 

Now  by  drawing,  or  imagining  to  be  drawn,  a  vertical 
line  upon  the  paper,  and  then  drawing  a  line  making  with 
it  an  angle  equal  to  BAC,  we  shall  have  the  direction  of 
AC,  or  its  inclination  to  a  plumb-line  AB. 

To  find  the  direction  of  a 
line  nearly  horizontal,  we 
have  but  to  balance  the  ru- 
ler, by  placing  its  centre 
upon  the  thumb  ;  then,  con- 
tinuing it  in  a  horizontal 
position,  and  bringing  it  to 
lange  with  the  point  A,  we 
discover  how  much  AB  va- 
ries from  a  true  level.  Con- 
ceiving or  drawing  such  an 
auxiliary  level  line  upon  the  paper,  and  then  laying  down 
the  angle  CAB,  we  shall  have  the  direction  of  AB,  or  its 
inclination  to  a  horizontal  line.     This  method  is  applicable 


TOPOGRAPHICAL    DRAWING.  63 

to  the  lines  of  distant  objects,  as  well  as  to  those  which 
are  near. 

Having  acquired  by  practice  the  power  of  sketching  a 
single  object  in  outline,  the  learner  should  place  two  or 
more  objects  before  him,  and  endeavor,  by  means  of  draw- 
ing their  outlines,  to  represent,  in  addition  to  their  forms, 
their  relative  position  with  respect  to  each  other.  He  should 
then  proceed  to  shade  them,  and  to  draw  the  shadows  which 
they  cast  upon  each  other,  and  upon  the  table  or  other  sur- 
face on  which  they  may  be  placed.  The  colors  of  the 
lights,  shades,  and  shadows  may  then  be  added,  and  the 
representation  will  be  complete. 


SECTION  II. 


TOPOGRAPHICAL    DRAWING. 


1.   What  is  Topographical  Drawing  ? 

Topographical  Drawing  is  the  art  of  representing  upon 
a  plane  surface,  the  character  and  features  of  any  pieco 
of  ground.  Such  drawings  are  always  plans,  and  are  dis- 
tinguished from  geographical  maps  by  a  greater  degree  of 
minuteness  in  their  details.  A  system  of  signs  has  been 
universally  agreed  upon,  and  adopted ;  most  of  which,  how- 
ever, have  a  sufficient  resemblance  to  the  objects  for  which 
they  stand,  to  enable  them  to  be  easily  recognised. 

The  signs  in  the  annexed  plates  have  been  adopted  by 
the  Engineer  Department,  and  are  used  in  all  the  plans 
and  maps  made  by  the  U.  S.  Engineers. 

These  we  shall  proceed  to  explain,  giving  at  the  same 
time  such  hints  as  to  the  manner  of  drawing  them,  as  may 
appear  to  be  necessary. 


64  BOOK    III. — SECTION    II. 

The  dimensions  in  which  we  represent  such  objects  as 
houses,  trees,  roads,  &c,  in  a  topographical  plan,  depend, 
of  course,  upon  the  scale  to  which  the  drawing  is  made. 
Generally,  for  the  sake  of  greater  distinctness,  they  are  en- 
larged to  two  or  three  times  their  proportionate  size  :  un- 
less the  scale  is  very  large,  or  when  one  of  the  objects  of 
the  plan  is  to  exhibit  every  thing  in  its  just  proportion. 

2.  Explain  the  figures  on  the  next  page. 

The  figures  in  the  first  column  explain  themselves,  in 
most  cases,  by  some  resemblance  or  appropriate  sign ;  in 
other  cases,  they  are  purely  conventional. 

In  Fig.  2,  the  signs  of  the  plants  are  placed  on  the  cor- 
ners of  squares  drawn  through  the  fields  they  occupy. 

Fig.  3  shows  the  manner  of  expressing  a  pine  forest  with 
roads  and  the  details  of  the  leaves,  in  case  the  scale  of  the 
drawing  will  admit  of  their  use.  In  forests,  the  trees  are 
placed  without  any  particular  order  or  arrangement. 

In  Fig.  4,  the  horizontal  lines,  or  the  lines  parallel  to  the 
top  and  bottom  of  the  drawing,  represent  the  watery  portion 
of  a  fresh-water  marsh :  the  rest  of  the  figure,  the  earthy  or 
grassy  parts.  In  general,  stagnant  water  is  represented  by 
horizontal  lines ;  and  meadow,  or  heath,  by  small  tufts  of 
grass.  The  combination  of  these  two  signs  indicates  mo- 
rass, or  wet  ground. 

Fig.  5  represents  hillocks,  or  sloping  ground.  The  paper 
is  always  left  white  to  denote  a  level;  and  each  one  of 
the  broken  lines  drawn  from  the  summit  to  the  base  of  a 
hill,  indicates  throughout  its  length  the  direction  of  the  slope, 
or  the  line  of  greatest  descent.  The  degree  of  bj^ckness, 
or  shade,  produced  by  these  lines  shows  the  nature  of  the 
slope,  from  the  perfect  white  of  a  level,  to  the  deep  black- 
ness of  an  almost  perpendicular  descent. 


i  &&&AJLA& 
AJLAJLJL&Jz 

AJL<AAA>A>A 


Fig.  2. 

i   f  f   f   f    f    f  f  i 

I  f  1»  tf  <F  f  <P  'F  j 

i  <p  fl  i]j  rp  1)  f  <f  i 

I  tf  f  A  H  <p  fp  D  I 

!  1»  fl   fl  #  A  tf  A  ! 

|  f  'P  <f  1>   'P  tftf  i 

iff    tf    *    f    *    tf  " 

L  "S 


Telegraph 


....im 


Fig.  4. 


Post-office 

House     


Fiff.  5. 


66  BOOK    III. — SECTION    II. 

3.  Explain  the  figures  on  the  next  page. 

Fig.  1  represents  a  rice-plantation ;  Fig.  2,  an  ornamen 
tal  garden ;  Fig.  3,  a  cotton-field ;  Fig.  4,  ploughed  land , 
Fig.  5,  an  orchard;  and  Fig.  6,  a  vineyard.  Figs.  1,  3,  5, 
and  6,  are  drawn  as  was  described  in  the  case  of  page  65, 
Fig.  2.  Where  it  is  not  necessary  to  describe  minutely  the 
kind  of  crop  existing  upon  the  land,  every  kind  of  cultiva- 
tion may  be  expressed  as  is  done  in  Fig.  4. 

Figs.  7,  8,  and  9  indicate,  respectively,  the  details  of  the 
leaves  for  oak,  fruit,  and  chestnut  trees,  whenever  their  use 
in  a  plan  is  desirable. 

Fig.  10  represents  a. heath  and  common  road.  It  is  left 
white,  being  a  level,  with  the  exception  of  the  tufts  of 
grass. 

Fig.  11  is  an  oak,  &c.  forest. 

Fig.  12  is  a  salt  marsh.  This  is  drawn  in  a  different 
manner  from  a  fresh-water  marsh,  being  composed  of  un- 
broken Horizontal  lines,  with  tufts  of  grass  interspersed 
among  them. 

Fig.  13  represents  meadow,  or  bottom  land,  with  a  small 
stream  running  through  it.  The  sign  for  the  grass  is  here 
more  regularly  disposed  than  in  a  heath,  or  common. 

Fig.  14  shows  the  mode  of  indicating  different  kinds  of 
roads,  fences,  paths,  &c. 

4.  How  is  water  represented! 

Running  water,  the  water  of  lakes,  and  water  that  is 
affected  by  tides,  are  always  represented  by  lines  drawn 
within  the  outline,  and  parallel  to  the  shores,  in  such  a 
manner,  that  by  gradually  increasing  the  distance  between 
the  lines,  which  are  at  first  very  close  together,  the  shade 
may  be  uniformly  lightened  from  the  shores  to  the  middle. 
The  course  of  the  current  is  indicated  by  an  arrow,  with 
the  head  turned  in  the  direction  in  which  the  water  runs. 


Fig 

1. 

!■£> 

£-$! 

;<I'V\ 

V'V  : 

#•  ''"•■/ 

I^M/I 

Fie. 


Fig.  3. 


■: 


Fig.  4. 


lit 

Fig.  7. 


Fig.  5. 

<k  *  ^  ^  ^  ^  ^ 
4  ^  <%  %,  %  ^  4- 

%  ^  %  <\  «*  <3k  4 

<k  <*.  ^  ^  \  ^  i. 

Fig.  8. 


Fig.  6. 

!  i  i  i  v  i  *  t ! 

iUHUl! 

h  tn  vHl 
I  $  s  s  %  %  s  s  \ 
i  \  \  \  \  \  \  i  \ 
Li  i  %  i  \  i  i  j 
i  i  $  \  s  i  *  t  i 

68  BOOK    III. SECTION    II. 

5.  Explain  the  figures  on  the  next  page. ' 

Fig.  1  represents  the  rocky  shore  ot  water  thus  shaded. 

Fig.  2  denotes  rocks  that  are  above  the  surface  of  the 
water.  Here,  also,  the  lines  indicate  the  direction  of  the 
descent  from  the  highest  point,  near  the  middle,  to  the 
water  line. 

Fig.  3  shows  the  manner  of  representing  salt-works. 

Figs.  4,  5,  and  6  show  the  three  conditions  of  sand-shoals. 

Fig.  7  is  a  sign  used  to  show  the  direction  of  the  current. 

Fig.  8  shows  that  there  is  no  current. 

Fig.  9  indicates  the  different  stages  of  the  tides  by 
means  of  dots  introduced  in  all  shading  above  low-water 
mark. 

Fig.  10  represents  rocks  sometimes  bare,  and  Fig.  11, 
sunken  rocks. 

Fig.  12  is  a  shore  with  sand-hillocks  and  fisheries. 

Fig.  13  is  a  collection  of  signs  used  for  describing  tho 
facilities  or  dangers  of  navigation. 

Fig.  14  exhibits  a  river,  with  the  different  circumstances 
connected  with  its  navigation,  and  the  means  of  crossing  it. 

Fig.  15  is  a  lake,  shaded  in  the  manner  before  de- 
scribed. 

In  shading  a  piece  of  water  by  this  method,  this  rule  must 
be  observed.  Having  drawn  the  outline,  conduct  the  first 
shading  line  along  every  shore,  (if  there  be  more  than  one,) 
and  around  all  islands,  keeping  it  as  clsse  as  possible  to 
the  shore-line. 

When  the  first  shading  line  is  thus  applied  everywhere, 
take  up  the  second  one,  laying  it  nearly  as  close  to  the 
first  as  the  first  is  to  the  outline.  When  the  second  line  is 
drawn  wherever  it  can  go,  take  up  the  third;  increasing 
gradually  and  uniformly  the  distance  between  the  lines,  un 


Fig.  7. 


Fig.  8. 


Fig.  9. 


Fig.  10. 


Fig.  12. 


70  BOOK    111. SECTION    III. 

til  they  approach  the  middle,  when  it  may  be  increased  a 
little  more  rapidly,  and  the  lines  made  somewhat  thinner. 

By  pursuing  this  system,  the  shade  will  be  graduated  in 
a  similar  manner  from  every  shore,  and  perfect  symmetry 
in  the  positions  of  the  lines  will  be  insured. 


SECTION   III. 

PRINCIPLES    OF    PLAN    DRAWING. 

1.  What  are  Geometrical  Drawings? 

Geometrical  drawings  are  those  which  are  made  for  the 
purpose  of  conveying  to  the  mind,  through  the  eye,  a  just 
idea  of  the  true  proportions  and  dimensions  of  objects. 

2.  What  objects  are  generally  represented  in  geometrical 
? 


The  objects  represented  in  geometrical  drawings  are  gen- 
erally solid  bodies,  with  irregular  or  curved  surfaces,  such 
as  houses,  blocks  of  wood,  chairs,  tables,  &c. 

3.  Can  we  generally  conceive  of  their  shape  and  dimensions 
from  one  single  drawing  or  view? 

We  cannot.     For  instance,  if  we  place  ourselves  in  front 

of  a   house,  or  opposite  to  one  end  of  it,  or  if  we  stand 

behind  it,  or  look   down  upon  it  from  some   great  height, 

such  as  the  top  of  a  lofty  steeple,  we  shall  in  each  case 

have  a  different  view  of  it ;  so  that,  unless  we  take  different 

m     . 
drawings  of  it,  from  several  points,  it  will  not  be  possible 

to  convey  any  just  notion  of  its  general  appearance. 

4.  What  is  a  horizontal  plane  ? 

It  is  any  plane  parallel  to  the  water-level,  such  as  the 
level  ground,  the  floor  of  a  house,  &c. 


PRINCIPLES    OF    PLAN    DRAWING.  71 

5.  What  is  a  vertical  plane  ? 

It  is  a  plane  perpendicular  to  a  horizontal  plane ;  such 
as  the  front  or  ends  of  a  house,  or  the  face  of  a  vertical 
wall. 

6.  Hew  many  kinds  of  geometrical  drawings  are  necessary 
in  order  to  represent  the  form  and  dimensions  of  an  object  ? 

Three  kinds  only  are  necessary ;  viz.,  a  plan,  a  sec- 
tion, and  an  elevation. 

7.  What  is  a  plan  ? 

A  plan  of  an  object  merely  resembles  the  appearance 
which  it  would  present  to  the  eye,  when  viewed  from  a 
point  directly  above  it. 

In  order  to  illustrate  this  more  clearly,  let  us  proceed  to 
draw  the  plan  of  a  small  building 

In  commencing  a  building,  the  first  thing  necessary  is  to 
have  a  general  plan,  or  plan  of  the  foundation.  Let  us 
suppose  that  the  building  to  be  represented  is  a  cottage, 
with  a  door  and  window  only. 

First,  having  fixed  upon  the  scale 
on  which  the  drawing  is  to  be  made, 
say  30  feet  to  the  inch,  lay  off  the 
length  of  the  cottage  30  feet,  on  the 
line  ab,  and  the  width  24  feet,  on  ac, 
and  complete  the  rectangle  to  repre- 
sent the  exterior  dimensions  of  the  cottage  ;  that  is  to  say, 
its  length  and  breadth  from  out  to  out. 

Next,  lay  off  from  the  same  scale 
the  thickness  of  the  wall  from  a  to 
b,  and  from  a  to  c,  and  draw  the  in- 
terior rectangle,  having  its  sides  par- 


allel, respectively,  to   those    of  the  b  a 

outer  one.     This  will   represent  the  interior  faces  of  the 
wall. 


72  BOOK    III. SECTION    III. 

We  see  that  this  figure  has  nearly  the  same  appearance 
as  would  be  presented  by  the  foundations  of  a  small  build- 
ing, viewed  from  a  point  directly  over  them. 

Doors  and  windows  are  generally 
marked  in  a  ground  plan.  In  order 
to  distinguish  them  from  each  other, 
the  lines  of  the  foundation  walls, 
which  interfere  with  the  doors,  are 
rubbed  out.  The  doors  and  windows 
will  be  marked  accordingly." 

The  complete  plan  of  the  cottage  is  now  drawn.  It 
shows  the  size  of  the  room,  the  thickness  of  the  walls,  and 
the  width  and  position  of  the  door  and  window. 

By  means  of  a  plan,  drawn  according  to  a  scale,  it  would 
be  easy  to  lay  out  correctly,  the  foundations  of  a  building 
and  the  doors  and  windows  of  the  lower  story.  But  after 
building  a  few  courses,  we  should  be  obliged  to  stop  for 
want  of  further  directions,  because  the  plan  can  neither 
explain  the  height  of  the  doors  or  windows,  nor  the  height 
of  any  other  part  of  the  building. 

This  proves  what  has  already  been  stated,  viz.,  that  more 
than  one  kind  of  drawing  of  any  object  is  always  necessary 
in  order  to  explain  its  form  and  dimensions.  Before  pro- 
ceeding to  the  other  kinds  of  geometrical  drawings,  men- 
tioned above,  we  will  add  some  further  explanations  and 
observations  on  the  subject  of  plans. 

8.  The  plan  of  any  object  is  always  supposed  to  be  made 
on  a  horizontal  plane  or  dead  level.  The  necessity  of  fol- 
lowing this  rule  will  appear  from  the  following  considera- 
tions. 

Suppose  it  were  required  to  build  a  house  on  uneven 
ground,  such,  for  example,  as  the  side  of  a  hill.  Every  one 
knows  that  in  laying  out  the  foundation,  no  reliance  would 
be  put  on  any  oblique  measurements  made  along  the  slope, 


PRINCIPLES    OF    PLAN    DRAWING.  73 

but  that  all  the  measurements  would  have  .0  be  made  in 
horizontal  lines.  For  instance,  if  you  were  to  measure  30 
feet  obliquely,  along  the  side  of  the  hill,  for  the  breadth  of 
your  proposed  building,  it  would  still  be  necessary  to  lay 
ihe  first  floor  horizontally.  After  this  was  done,  you  might 
find  the  space  which  was  laid  out  for  the  breadth  of  the 
building,  reduced  to  29  feet,  to  28  feet,  to  25  feet,  or  even 
to  a  less  distance,  according  to  the  steepness  of  the  slope 
of  the  hill.  The  plan  of  an  uneven  field,  in  which  the  di- 
mensions were  marked  according  to  oblique  measurements 
made  upon  the  sloping  or  irregular  surface  of  the  ground, 
would  therefore  be  of  no  use. 

9.  It  is  more  difficult  to  draw  the  plan  of  any  object 
having  sloping  or  oblique  lines,  than  to  draw  the  plan  of  a 
building  having  only  horizontal  and  vertical  lines,  because 
the  oblique  or  sloping  lines  must  all  be  reduced  in  a  certain 
proportion. 

10.  The  following  are  the  rules  for  laying  down  truly,  on 
a  horizontal  plane,  the  points  and  lines  of  all  objects,  any 
way  situated,  with  respect  to  it. 

11.  The  imaginary  horizontal  plane,  on  which  the  plan 
is  made,  and  to  which  all  points  and  lines  are  referred,  is 
called  the  horizontal  plane  of  projection. 

This  plane  may  be  so  taken  as  to  cut  the  object  which 
is  to  be  drawn  upon  it,  or  it  may  be  taken  directly  above  or 
below  the  object.  But  for  learners,  it  is  best  to  begin  by 
supposing  the  horizontal  plane  to  pass  through  the  base, 
or  lowest  point  of  the  given  object. 

In  respect  to  such  points  of  the  object  as  stand  upon  the 
plane  of  projection,  or  coincide  with  it,  there  can  be  no 
difficulty,  for  such  points  are  their  own  place  or  projections 
on  the  plane. 

4 


74  BOOK    111. SECTION    111. 

From  every  point  without  the  plane  of  projection,  a  per- 
pendicular is  supposed  to  be  drawn  to  the  plane,  and  the 
point  in  which  this  perpendicular  pierces  the  plane,  will 
mark  the  true  position  of  the  point  from  which  it  was 
drawn. 

If  the  plane  of  projection  be  supposed  to  lie  below  the 
given  object,  then  all  the  points  of  the  object  will  be  above 
the  plane  of  projection;  and,  consequently,  all  the  perpen- 
diculars, requisite  for  finding  the  position  of  these  points 
on  the  plane  of  projection,  will  go  downward  from  these 
points. 

But  if  the  plane  of  projection  be  supposed  to  be  above 
the  given  object,  then  the  several  points  of  the  object  will 
be  below  the  plane  ;  and,  consequently,  all  the  perpendicu- 
lars, necessary  for  finding  the  position  of  these  points  on 
the  plane  of  projection,  will  slope  up  from  the  given 
points. 

12.  Since  the  plane  of  projection,  in  plans,  is  always 
supposed  to  be  horizontal,  every  perpendicular,  whether 
dropped  or  raised,  will  be  a  vertical  or  plumb-line.  Con- 
sequently, if  we  suppose  two  plummets  to  be  suspended 
exactly  over  two  points  of  an  object,  the  plan  of  which  is 
required  to  be  drawn,  the  distance  between  the  plumb-lines, 
measured  perpendicularly,  will  be  the  true  distance  at  which 
the  two  points  ought  to  be  laid  down  on  the  plan. 

13.  To  explain  this,  draw 
three  lines  on  the  board  con- 
nected together ;  all  of  the 
same  length,  but  sloping  un- 
equally. These  may  repre- 
sent the  form  of  some  sloping 
or  oblique  object,  of  which  the 
plan  is  to  be  drawn. 


PRINCIPLES    OF    PLAN    DRAWING. 


75 


The  pupils  will  copy  this 
and  the  following  operations 
on  their  slates,  without  fur- 
ther directions,  until  the  figure 
is  completed.  From  the  ex 
tremities  of  each  of  the  three 
lines,  draw  dotted  lines,  paral- 
lel to  each  other,  directed  to- 
wards the  top  of  the  board. 

These  dotted  lines  may  rep- 
resent plumb-lines  held  over 
the  various  points  of  the  ob- 
lique object.  Now  mark  the 
various  points  of  the  oblique 
object  by  capital  letters  A,  B, 
C,  and  D,  from  left  to  right. 

As  the  distances  between 
the  four  plumb-lines,  repre- 
sented in  the  last  figure,  must 
be  meastrred  perpendicularly, 
not  obliquely,  draw  a  line 
above  the  given  object,  and 
perpendicular  to  the  dotted 
lines,  on  which  the  said  distances  are  to  be  measured. 

At  the  points  where  the  perpendiculars  meet  the  horizontal 
line,  make  the  letters  a,  b,  c,  and  d,  from  left  to  right. 

The  distance  between  the 
points  a  and  b,  at  the  top  of 
the  figure,  represents  the  exact 
distance  between  the  plumb- 
lines  suspended  over  the  points 
A  and  B.  Consequently,  the 
perpendicular  line  ab,  at  the 
top  of  the  figure,  represents 


76 


BOOK    III. SECTION    III. 


the  exact  length  which  ought  to  be  given  to  the  oblique 
line  AB,  in  drawing  a  plan  of  the  given  object. 

The  perpendicular  line  be,  at  the  top  of  the  figure,  in  like 
manner,  "and  for  the  same  reason,  represents  the  exact  dis- 
tance which  ought  to  be  given  to  the  oblique  line  BC,  in 
the  plan  of  the  object. 

And  the  perpendicular  cd,  at  the  top  of  the  figure,  in  like 
manner  represents  the   exact  distance  which  ought  to  be 

given  to  the  oblique  line  CD,  in  the  plan. 

a         b    •  c  d 

14.  Let  us  now  produce  the 
dotted  lines  below  the  given 
object,  and  draw  a  second  ho- 
rizontal line  intersecting  them 
perpendicularly ;  and  let  us 
also  mark  the  points  of  inter- 
section  by  the   same    letters 

a,  b,  c,  and  d. 
Then,  since   parallel  lines 

are  always  at  the  same  distance  from  each  other,  although 
produced  ever  so  far,  the  distance  between  the  p6ints  a  and 

b,  at  the  bottom  of  the  figure,  will  be  equal  to  the  distance 
between  the  points  a  and  b  at  the  top ;  and  the  same  for  the 
distances  between  any  other  two  points. 

Consequently,  the  perpendicular  distances  ab,  be,  and  cd, 
at  the  bottom  of  the  figure,  will  be  equal  to  the  perpendicu- 
lar distances  ab,  be,  and  cd,  at  the  top ;  and,  therefore,  the 
lines  ab,  be,  and  cd,  at  the  bottom,  will  serve  equally  well 
to  represent  the  respective  lengths  which  ought  to  be  given 
to  the  oblique  lines  AB,  BC,  and  CD,  in  the  plan  of  the 
given  object. 

Hence  we  see,  that  either  the  upper  horizontal  line  ab, 
or  the  lower  horizontal  line  ab,  may  represent  the  plane  of 
projection,  to  be  used  in  drawing  the  plan  of  the  oblique 
object ;  the  upper  line  will  represent  a  plane  passing  above 


PRINCIPLES    OF    PLAN    DRAWING.  77 

the  given  object,  and  the  lower  line  a  plane  passing  be- 
low it. 

This  illustrates  what  was  before  observed,  that  in  draw- 
ing the  plan  of  any  object,  it  makes  no  difference  whether 
the  plane  of  projection  is  taken  above  or  below  it. 

15.  A  line  drawn  from  any  point  in  a  given  object,  and 
perpendicular  to  the  plane  of  projection,  is  called  the  pro- 
jecting line  of  the  point ;  and  the  place  where  the  perpen- 
dicular meets  the  plane,  is  called  the  projection  of  the  point. 

16.  Let  us  illustrate  the  above  rules  by  means  of  a  square 
pyramid. — (Here  let  the  teacher  explain  the  shape  of  a 
square  pyramid,  and  exhibit  one  to  the  class.) 

If  we  look  down  upon  a  square  pyramid,  we  shall  sea 
the  extremities  of  its  base,  its  vertex,  and  the  four  edges 
or  oblique  lines  which  are  formed  by  the  meeting  of  its 
sides.  All  these  particulars  must  therefore  be  represented 
in  the  plan  of  a  square  pyramid. 

The  most  convenient  way,  is  to  suppose  the  horizontal 
plane  on  which  the  plan  is  to  be  made,  to  pass  through  the 
base  of  the  pyramid.  For  example,  if  we  place  the  pyra- 
mid upon  a  table,  the  level  surface  of  the  table  will  repre 
sent  the  plane  of  projection.  The  base  of  the  pyramid, 
now  standing  on  the  plane  of  projection,  coincides  with  it, 
and  will  be  its  own  projection,  without  any  enlargement  or 
diminution. 

The  base  of  the  pyramid  is  a 
square.  Represent  it,  therefore,  on 
the  paper  or  board,  by  drawing  a 
square  exactly  equal  to  it. 

In  the  present  instance,  the  base 
of  the  pyramid  coinciding  with  the 
plane  of  projection,  and  the  pyramid 
being  perfectly  regular,  it  is  evident 


78  BOOK    111. SECTION    III. 

that  a  perpendicular  dropped  from  the  vertex,  would  fall 
exactly  on  the  middle  or  central  point  of  the  base.  Mark, 
therefore,  the  middle  point  of  the  square,  and  it  will  repre- 
sent the  projection  of  the  vertex  of  the  pyramid. 

The  four  ridges,  or  oblique  lines,  remain  to  be  drawn. 
But,  one  extremity  of  each  of  these  lines  passes  through 
each  angular  point  of  the  base,  all  of  which  are  already 
marked  on  the  plan.  The  other  extremities  of  these  lines 
all  meet  at  the  vertex  of  the  pyramid,  whose  projection  on 
the  plan  is  also  determined.  There- 
fore, draw  from  the  centre  of  the 
base  four  straight  lines,  one  to  each 
angle  of  the  base,  and  they  will 
represent,  in  the  plan,  the  four  ridges 
of  the  pyramid. 

You  see  that  the  plan  of  the  pyra- 
mid, now  drawn  upon  paper,  shows 
no  dimensions  but  those  of  the  base. 

It  also  indicates  the  particular  point  of  the  base  over  which 
the  vertex  stands  ;  but  it  neither  explains  the  height  of  the 
pyramid,  nor  the  obliquity  or  slopes  of  its  sides. 

The  plan,  therefore,  cannot  alone  explain  the  nature 
either  of  a  building  or  pyramid,  or  of  any  other  object,  and 
recourse  must  be  had  to  some  other  kind  of  drawings. 

of  sections.  • 

17.  A  section  is  a  plane  figure,  formed  by  cutting  any 
solid  body  into  two  parts.  A  solid  body  may  be  cut  in  a 
groat  number  of  directions  :  viz.,  horizontally,  vertically,  and 
obliquely:  and  hence,  the  number  of  sections  which  may 
be  formed  of  any  object,  are  infinite,  or  beyond  calcula- 
tion. 

To  avoid  the  confusion  which  might  arise  in  plan-draw- 
ing, from  sections  taken  at  random,  the  geometrical  drawing 


OF    SECTIONS.  79 

called  a  section,  is  always  taken  vertically;  that  is  to  say, 
the  object  is  supposed  to  be  cut,  right  down,  perpendicularly, 
from  top  to  bottom,  by  a  vertical  plane  ;  in  other  words,  it 
is  supposed  to  be  cut  everywhere  in  a  plumb-line. 

A  section  is  principally  intended  to  snow  the  heights  of 
objects,  and  thereby  to  make  up  for  the  defects  of  the  plan, 
which  have  already  been  explained. 

Supposing  it  were  required  to  measure  the  height  of  one 
of  the  sides  of  a  room.  This  could  not  be  correctly  done 
by  measuring  diagonally  or  obliquely — that  would  be  quite 
wrong.  There  is  no  way  of  finding  the  true  height  except 
by  measuring  vertically,  or  in  the  plumb-line. 

If,  then,  we  suppose  a  section  of  the  room  to  be  taken 
in  which  we  now  are,  it  is  evident  that  if  the  section  were 
taken  in  a  sloping  direction,  it  would  cut  the  sides  of  the 
room  obliquely.  Such  a  section  would  therefore  give  an 
erroneous  representation  of  the  sides  of  the  room. 

Sections  taken  across  any  building  or  object,  will  of 
course  serve  to  show  the  breadth  as  well  as  the  height  of 
its  various  parts.  In  order  that  this  may  be  done  truly, 
another  rule  must  be  laid  down  no  less  essential  than  the 
former;  viz., 

In  taking  the  section  of  any  regular  object,  such  as  a 
rectangular  building,  the  object  is  always  supposed  to  be 
cut  right  across ;  that  is,  in  a  direction  perpendicular  to 
two  opposite  sides  ;  and  the  same  reason  holds  good  in  this 
case,  which  was  given  for  employing  a  vertical  plane. 

Supposing  we  wished  to  measure  the  breadth  of  this 
room.  You  see  at  once,  that  if  we  took  the  measurement 
obliquely,  from  angle  to  angle,  the  result  would  be  quite 
wrong ;  and  that  there  is  no  possible  way  of  measuring  the 
breadth  of  the  room  accurately,  except  in  a  direction  per- 
pendicular to  its  two  opposite  sides. 

From  these   considerations  it  must  be  evident,  that  any 


80 


BOOK    III. SECtlON    III. 


section  of  a  building,  or  of  an  object,  taken  in  a  sloping  01 
oblique  direction,  would  not  be  of  the  smallest  use,  because 
it  would  cither  misrepresent  the  height,  or  the  bieadth,  or 
both. 


\b  * 


18.  This  being  premised,  let  us  now  pioceed  to  draw 
a  section  of  the  small  cottage,  of  which  we  have  already 
drawn  the  plan. 

Let  us  suppose  that  the  proposed  section  is  required  to 
pass  through  the  door  of  the  building.  Draw  a  dotted  line 
perpendicularly  across  .  the  plan 
of  the  cottage,  passing  through 
the  door.  This  dotted  line  will 
represent  the  direction  in  which 
the  proposed  section  is  to  be 
taken.  a 

Mark  the  points  on  the  plan  where  the  dotted  line  cuts  the 
front  and  back  walls  of  the  cottage,  by  the  letters  o,  b,  c, 
and  d.  The  distances  between  the  points  a,  b,  c,  and  d,  show 
the  breadth  of  the  cottage  and  the  thickness  of  the  walls. 

As  the  same  dimensions  which  have  been  used  in  the 
plan  must  be  again  represented  in  the  section,  it  will  save 
time  to  transfer  the  whole  of  them,  at  once,  from  the  plan 
to  the  section. 

Therefore,  draw  a  separate  line  to  represent  the  level 
of  the  building,  which  will  also  be  the  ground  line  or  base 

d 


\c 
j 

Jb 


a  abed 

of  the  section.     Then  divide  this  line  in  the  same  manner 
as  the  dotted  line  abed  is  divided  in  the  plan. 


PRINCIPLES    OF    PLAN    DRAWING. 


81 


Under  the  respective  points  of  division,  on  this  new  line, 
mark  the  same  letters  a,  b,  c,  and  d.  When  this  is  done, 
the  corresponding  or  like  parts  of  both  lines  will  be  known 
by  inspection. 

From  the  points  a,  b,  c,  and  d,  on  the 
ground  line  of  the  section,  which  rep- 
resents the  position  and  thickness  of  the 
walls  of  the  cottage,  raise  perpendiculars 
to  show  the  height  of  the  walls.  Join' 
the  tops  of  these  perpendiculars  by  a  dotted  line  which  will 
be  horizontal,  and  this  line  will  show  the  level  from  whence 
the  roof  is  supposed  to  spring. 

The  plan  of  the  cottage  is  still  supposed  to  remain  on  the 
board  and  slates,  but  is  left  out  in  some  of  the  following 
figures.  It  will  again  be  occasionally  introduced,  whenever 
it  shall  be  necessary  to  point  out  the  connection  between  the 
plan  and  the  section.  t 


19.  We  will  now  suppose  the  roof  to  be 
a  regular  pitch  roof.  Therefore,  bisect 
the  last-drawn  line,  in  order  to  find  the 
middle  of  the  building ;  and  from  the  point 
of  bisection  raise  a  perpendicular,  to  show 
the  height  of  the  roof.  From  the  extremi- 
ties of  this  perpendicular,  draw  an  oblique_ 
line  to  the  outside  of  the  top  of  each  wall : 
this  will  show  the  sides  of  the  roof.  Then  draw  right  lines 
interiorly,  parallel  to  the  last  lines,  to  show  the  thickness 
of  the  roof. 

As  the  section  is  supposed  to  pass  through  the  door  of 
the  cottage,  a  line  must  be  drawn  to  represent  the  top  of 
the  door,  and  to  show  the  height  of  it. 

The  section  which  has  just  been  drawn,  is  only  intended 
to  give  a  general  notion  of  this  kind  of  geometrical  drawing. 

4* 


82 


BOOK    III. SECTION    III. 


Many  particulars  are  therefore  omitted,  which  it  would  be 
proper  to  introduce  into  a  finished  section  of  a  building. 
For  instance,  the  depth  and  thickness  of  the  foundation, 
the  recess  of  the  door,  the  thickness  of  the  rafters  and  other 
parts  of  the  roof;  also,  its  projection  over  the  walls,  if 
formed  with  eaves.  These,  and  other  details  might  easily 
have  been  represented,  by  adding  a  few  more  lines.     The 


d 

\c 

{         Plan. 

\b 

II           II 

a  b  c  d 

rough  section  of  the  cottage  is  now  complete,  and  you  may 
observe,  that  those  dimensions  which  are  marked  with  the 
same  letters,  agree  in  both. 

The  plan  and  sections,  as  they  stand  at  present,  explain 
sufficiently  the  general  dimensions  of  the  cottage,  and  the 
proportions  of  the  roof  and  door ;  but  they  do  not  show  the 
height  of  the  windows,  nor  the  general  appearance  of  the 
building. 

The  latter  particulars  cannot  be  represented  without  the 
assistance  of  the  third  kind  of  geometrical  drawing,  before 
mentioned,  called  an  elevation. 


OF    THE    ELEVATION. 

20.  An  elevation  is  the  view  of  any  upright  side  of  a 
building  or  other  object,  nearly  such  as  it  would  appear 
to  a  person  standing  directly  in  front  of  it. 

In  order  to  understand  this  definition  more  clearly,  let 
us  draw  an  elevation  of  the  front  of  the  cottage. 


OF    THE    ELEVATION. 


83 


12    3         4    5       6 


As  the  principal  dimensions  of 
the  front  of  the  cottage  appear  in 
the  plan,  let  the  various  points  be 
marked  by  the  figures  1,  2,  3,  4,  5, 
and  6. 

The  points  thus  marked  show 
the  length  of  the  front  of  the  cot- 
tage,  and  the  breadth  and  position  of  the  door  and  win- 
dow. 

As  all  these  dimensions  must  appear  in  the  elevation  of 
the  cottage,  the  easiest  method  will  be  to  transfer  them 
from  the  plan  to  the  elevation  at  once. 

Draw,  therefore,  a  separate  line,  to  represent  the  ground 
line,  or  level  upon  which  the  front  of  the  cottage  stands  ;  and 
upon  this  line,  set  off  a  distance  equal  to  the  length  of  the 
cottage,  and  divide  it  in  the  same  manner  as  the  front  of 
the  cottage  is  divided  in  the  plan. 

Mark  also  the  various  points  of  division  on  this  new  line, 
by  the  figures  1,  2,  3,  4,  5,  and  6.     When  this  is  done,  the 


_~   T 


1        2  3 


4    5 


2  3 


4  5       6 


corresponding  or  equal  parts  in  the  plan  and  in  the  ground 
line  of  the  elevation,  are  known  by  inspection.  From  the 
points  1  and  6  of  the  ground  line  of  the  elevation,  let  per- 
pendiculars be  drawn  to  show  the  height  of  the  walls.  Now, 
since  the  height  of  the  walls  is  already  represented  in  the 
section,  take  that  height  in  the  dividers  and  lay  it  off  on 
the  perpendiculars  through  1  and  6. 


84 


BOOK    III. —  SECTION    III. 


Join  the  top  of  these  perpendiculars  by  a  straight  line, 
This  line  will  represent  the  bottom  of  the  roof  of  the  cottage . 


123         456  123         456 

From  the  points  2  and  3  of  the  ground  line  of  the  eleva- 
tion, which  represent  the  width  of  the  door,  raise  perpen- 
diculars to  show  the  height  of  the  door.  Find  the  proper 
length  of  these  perpendiculars  by  measuring  the  height  of 


123         456  123         456 

the  door  in  the  section,  and  then  transfer  it  to  the  elevation. 
Complete  the  form  of  the  door  by  joining  the  top  of  the 
above  perpendiculars. 

From  the  points  4  and  5  in  the  elevation,  which  mark 
the  position  of  the  window,  raise  perpendiculars  to  find  the 
sides  of  the  window.     Next  complete  the  window  by  draw- 


ing the  top  and  bottom  of  it,  at  their  proper  height.  Dot 
that  part  of  each  of  the  last  perpendiculars,  which  falls 
below  the  bottoms  of  your  windows. 


OF    THE    ELEVATION. 


65 


The  form  of  the  roof  is  now  alone  wanting.  The  length 
of  the  roof  must  of  course  be  equal  to  the  length  of  the 
building,  and  the  height  of  it  may  be  found  by  referring  to 
the  section. 

It  is  a  general  rule,  in  geometrical  elevations,  never  to 
represent  the  height  of  any  sloping  object  by  oblique  meas- 
urements taken  along  the  slope  ;  but,  by'  dropping  a  perpen- 
dicular from  the-  highest  point,  or  vertex  of  the  slope,  to  the 
level  of  the  lowest  point  or  base  of  it. 

In  short,  the  height  of  any  sloping  object  in  a  geometrical 
elevation  is  measured  by  that  perpendicular  line,  which 
would  be  called  the  altitude  of  any  similar  figure  or  body 
in  Geometry.  Therefore,  in  transferring  the  height  of  the 
roof  from  the  section  to  the  elevation,  make  it  in  the  eleva* 
tion  equal  to  the  dotted  perpendicular  drawn  in  the  section. 
Next  draw  the  roofs :  when  this  is  done,  the  drawings  of 
the  cottage  are  as  follows : 


HI 


Plan. 


Section. 


Elevation. 


21.  The  Plan,  Section,  and  Elevation  of  a  small  cottage 
are  noAV  complete,  and  from  these  three  geometrical  draw- 
ings put  together,  every  dimension  necessary  for  explaining 
the  proportions  of  the  building  may  be  known. 

The  length  of  the  building  is  shown  in  the  plan  and  ele- 
vation, and  is  the  same  in  both. 

The  breadth  of  the  building,  and  thickness  of  the  walls, 


96  BOOK    III. SECTION    III. 

are  shown  in  the  plan  and  section,  and  are  the  same  in 
both. 

The  breadth  of  the  door,  and  that  of  the  window,  are 
shown  in  the  plan  and  elevation. 

The  height  of  the  door  is  shown  in  the  section  and  ele- 
vation, and  is  the  same  in  both. 

The  height  of  the  window  is  shown  in  the  elevation  only. 
But  if  the  section  had  been  taken  through  the  window,  in- 
stead of  the  door,  then  the  height  of  the  window  would 
have  been  shown,  and  not  that  of  the  door. 

The  height  of  the  walls,  and  the  perpendicular  height 
of  the  roof,  are  shown  in  the  section  and  elevation,  and 
are  equal  in  both.  But  jhe  particular  form  of  the  roof  is 
clearly  explained  in  the  section  only. 

REMARKS    ON    ELEVATIONS. 

22.  An  elevation  is  always  supposed  to  be  drawn  on  a 
vertical  plane,  which  is  called  the  vertical  plane  of  projec- 
tion. 

Those  points  of  an  object  which  lie  in,  or  coincide  with, 
the  plane  on  which  the  elevation  is  drawn,  are  their  own 
projections  on  that  plane.  Those  points  of  the  given  object 
which  lie  without  the  plane  of  projection,  must  be  trans- 
ferred to  it,  by  lines  drawn  from  the  points  and  perpendicular 
to  the  plane  of  projection.  Such  lines  are  called  projecting 
lines. 

Since  all  the  projecting  lines  which  determine  an  eleva- 
tion are  perpendicular  to  a  vertical  plane,  they  must  neces- 
sarily be  horizontal.  The  walls  or  sides  of  a  building  are 
vertical  planes,  being  built  according  to  a  plumb-line  ;  and 
therefore,  in  taking  a  geometrical  elevation,  the  plane  on 
which  it  is  made  may  be  supposed  to  coincide  with  the  front 
of  the  building,  or  any  other  side  which  is  to  be  repre- 
sented 


REMARKS    ON    ELEVATIONS.  87 

When  this  is  done,  the  length  and  height  of  the  side  of 
the  building,  and  the  neight  and  breadth  of  the  doors  and 
windows,  &c,  may  be  laid  down  in  a  geometrical  elevation, 
according  to  their  actual  dimensions  from  measurement. 

The  roof,  from  its  sloping  figure,  is  the  only  part  of  the 
exterior  side  of  the  building  which  cannot  agree  with  the 
plane  of  projection ;  and  hence,  in  drawing  the  elevation 
of  the  cottage,  it  was  necessary  to  diminish  the  oblique 
lines  of  the  slope  of  the  roof,  in  order  to  find  the  true  ver- 
tical height  of  it.  They  were  diminished  in  the  same  way 
that  the  oblique  lines  are  diminished  in  a  plan,  in  order  to 
find  the  base  of  any  slope. 

23.  It  is  not  necessary,  in  a  geometrical  elevation,  that 
the  plane  of  projection  should  be  supposed  to  agree  ex- 
actly with  the  upright  side  of  the  building  or  object  which 
is  to  be  represented.  But  when  they  do  not  agree,  it  is 
necessary  that  the  plane  of  projection  should  be  parallel 
to  the  upright  side  of  the  building  or  object,  of  which  the 
elevation  is  to  be  drawn.  In  that  case,  the  projecting  per- 
pendicular will  form,  on  the  plane  of  projection,  a  figure 
exactly  similar  to  the  front  of  the  building  or  object.  Con- 
sequently, if  you  suppose  a  plane  of  projection  to  be  chosen, 
parallel  to  the  upright  side  of  a  building  or  other  object,  of 
which  an  elevation  is  required,  then  the  dimensions  of  the 
various  parts  of  the  upright  side  of  the  given  object  may 
be  laid  down  in  the  drawing  in  their  true  proportions,  ac- 
cording to  measurement. 

24.  From  the  figures  which  have  been  drawn,  and  the 
instructions  which  have  been  given,  on  the  subject  of  Plan- 
drawing,  it  appears  that  plans  and  elevations  are  drawn 
exactly  according  to  the  same  principles,  with  only  this 
difference :  that  in  a  plan,  the  plane  of  projection  is  always 


88  BOOK    III. SECTION     III. 

horizontal,  whereas,  in  an  elevation,  n  is  always  verti- 
cal 

All  horizontal  planes,  which  may  be  used  as  planes  of 
projection  for  drawing  the  plan  of  a  building,  will  be  parallel 
to  each  other ;  but  the  vertical  planes,  on  which  the  ele- 
vations are  drawn,  may  be  oblique,  perpendicular,  or  paral- 
lel to  each  other. 

For  example,  the  several  floors  of  any  building,  being  all 
level,  and  all  the  points  of  each  at  the  same  height  from 
the  ground,  are  horizontal  planes  parallel  to  each  other. 
But  of  the  walls  of  a  building,  which  are  all  vertical  planes, 
some  two  of  them  may  be  perpendicular  to  each  other,  such 
as  the  side  and  end  walls ;  while  others  may  be  oblique  to 
each  other,  as  is  often  seen  in  irregular  buildings. 

OF    OBLIQUE    ELEVATIONS. 

25.  If,  in  drawing  the  elevation  of  any  rectangular  build 
ing,  the  plane  of  projection  were  chosen  oblique  to  one  of 
the  sides,  instead  of  parallel  to  it ;  then,  the  length  of  that 
side  of  the  building  and  the  breadth  of  the  doors  and  win- 
dows would  be  diminished  in  the  drawing,  in  such  a  manner 
as  to  give  a  false  notion  of  the  object.  In  an  oblique  eleva- 
tion of  this  kind,  the  projecting  lines  which  are  drawn  per- 
pendicular to  the  plane  of  projection,  will  be  oblique  to  the 
building ;  and  hence,  all  the  dimensions  except  those  which 
are  vertical  would  be  diminished  or  misrepresented  in  the 
drawing:  hence,  such  elevations  are  of  little  use,  and  are 
therefore  seldom  made. 

But,  although  oblique  elevations  of  the  fronts  of  buildings 
are  seldom  made,  it  often  happens  that  the  front  of  a  fine 
building  is  ornamented  with  columns,  mouldings,  and  archi- 
tectural decorations,  many  parts  of  which  are  oblique  to  the 
general  plane  of  the  front  of  the  building,  beyond  which 
they  project. 


OF    OBLIQUE    ELEVATIONS. 


89 


The  proper '  methods  of  representing  such  ornaments  in 
geometrical  elevations,  cannot  therefore  be  well  understood, 
unless  the  principle,  according  to  which  oblique  elevations 
of  any  upright  object  may  be  drawn,  is  clearly  explained. 

This  being  premised,  we  shall  give  an  example  of  the 
method  of  drawing  an  oblique  elevation  of  the  cottage,  of 
which  we  have  already  drawn  the  plan,  and  section,  and 
geometrical  elevation. 


26.  Resume  the  plan  before  drawn,  and  mark  thereon 
the  points  1,  2,  3,  4,  5,  and  6. 

A  straight  line  must  next  be 
drawn,  to  represent  the  new  plane 
of  projection,  on  which  the  oblique 
elevation  is  to  be  made.     This  new 

plane  of  projection  may  either  be  1  2  3  4  5  6 
supposed  to  coincide  with  some  line  of  the  front  face  of 
the  building  or  not.  We  shall  take  it  to  coincide  or  agree 
with  that  extremity  of  the  front  of  the  building  which  is 
marked  by  the  figure  6. 

Draw,  therefore,  a  right  line  through  the  point  6,  forming 
an  acute  angle  with  the  front  of  the  building,  and  this  line 
will  represent  the  new  plane  of  projection,  which  is  vertical. 

From  the  various  points  of 
the  front  of  the  building,  draw 
perpendiculars  to  the  last  line, 
and  dot  them.  These  perpen- 
diculars will  determine  the  true 


places  of  the  points  in  the  ob-     L L  U 

lique  elevation. 

Mark,  therefore,  in  like  man- 
ner, by  the  figures  1,  2,  3,  4,  5, 
and  6,  the  several  corresponding  points  on  that  line  which 
represents  the  plane  of  projection. 


90 


BOOK    III. — SECTION    III. 


That  end  of  the  cottage  which  is  nearest  to  the  plane  of 
projection  must  also  be  represented.  One  extremity  of  it 
coincides  with  the  said  plane.  From  the  other  extremity 
draw  a  dotted  perpendicular  to  the  plane  of  projection,  and 
mark  the  corresponding  points,  at  the  ends  of  this  line,  by 
the  figure  7. 

The  distance  between  the  points  1  and  6  in  the  plan, 
shows  the  length  of  the  front  of  the  cottage ;  and  there- 
fore the  distance  between  the  corresponding  points  1  and  6, 
on  the  plane  of  projection,  will  also  represent  the  length 
which  ought  to  be  given  to  the  front  of  the  cottage  in  the 
oblique  elevation. 

The  distance  between  the  points  6  and  7,  in  the  plan, 
represents  one  end  of  the  cottage ;  and  therefore  the  dis- 
tance between  the  corresponding  points  6  and  7,  on  the 
plane  of  projection,  will  also  represent  the  length  which 
ought  to  be  given  to  that  end  of  the  cottage  in  the  oblique 
elevation. 

And,  in  like  manner,  as  the  breadths  of  the  door  and  win- 
dow are  represented,  respectively,  by  a  certain  distance  in 
the  plan  ;  so  the  same  dimensions,  in  the  oblique  elevation, 
must  be  represented  by  the  distance  between  the  corre- 
sponding points,  on  the  plane  of  projection. 


H — \—+- 


-hh — ^- 


12   3       45    6       7 

You  will,  therefore,  draw  a  line  for  the  ground  line  of 
the  oblique  elevation.  Divide  this  line  in  the  same  manner 
as  the  one  which  represents  the  plane  of  projection,   and 


OF    OBLIQUE    ELEVATIONS. 


Ul 


2     3 


4     5 


'mark  it  with  the  same  numeral  figures.  From  the  points 
1,  6,  and  7,  on  the  ground  line 
of  the  elevation,  raise  perpen- 
diculars equal  to  the  height  of 
the  cottage  ;  and  draw  the  up- 
per line.  At  2  and  3  also 
draw  perpendiculars,  and  lay- 
off the  height  of  the  door ;  and 
do  the  same  at  4  and  5  for  the 

window — dotting  those  parts  of  the  perpendiculars  which  lie 
below  the  window. 

The  roof  only  remains  to  be  drawn.  Before  this  can  be 
done,  it  will  be  necessary  to  find  the  points  where  the  ridge 
ought  to  be  laid  down  in  the  plane  of  projection. 

The  ground  plan  of  the  cottage  does  not  show  the  ridge 
of  the  roof;  but  it  is  evident  that  the  ridge  of  a  regular  roof 
with  a  simple  pitch,  must  be  directly  over  the  middle  of  the 
building. 

In  order  to  save  the  trouble 
of    drawing   a    separate    plan, 
draw  a  dotted  line  RR  through  R 
the  middle  of  the  plan  already- 
drawn,  to  represent  the  ridge. 

From  the  points  R  and  R, 
which  represent  the  extremities 
of  the  ridge  of  the  roof,  draw 
dotted  perpendiculars  to  the 
plane  of  projection,  and  mark  R 
the  points  where  they  meet  the 
plane,  by  the  letters  R  and  R. 
The  points  R  and  R  must  next 
be  transferred  to  the  ground 
line  of  the  oblique  elevation. 
From  these  new  points  R  and  R,  draw  perpendiculars  and 


92 


BOOK    III. SECTION    III. 


lay  off  the  height  of  the  cottage,  which  is  found  by  referring 
to  the  section :  then  draw  the  upper  line,  which  will  repre- 
sent the  ridge ;  after  which,  draw  oblique  lines  from  the 
extremities  of  the  ridge  to  the  proper  points,  in  order  to 
complete  the  form  of  the  roof. 

The  oblique  elevation  of  the  cottage  is  now  finished,  as 
below,  where  the  parallel  elevation  is  also  given. 


1  R2  3  4   5       6  R  7 

The  heights  of  the  various  parts  of  the  oblique  elevation 
agree  with  those  of  the  section  and  front  elevation ;  but  all 
the  other  dimensions  are  changed,  being  less  than  they  were 
in  the  plan. 

If  the  plane  on  which  the  oblique  elevation  was  made 
had  formed  a  greater  angle  with  the  front  of  the  building, 
then  the  various  dimensions,  in  that  part  of  the  oblique  ele- 
vation which  represents  the  front  of  the  cottage,  would  have 
been  still  more  diminished. 

In  the  direct  elevation  of  the  front  of  the  cottage,  it  was 
not  necessary  to  take  any  notice  of  the  points  R  and  R, 
because  they  fell  directly  over  the  ends  of  the  building. 
The  two  ends  of  the  building  being  perpendicular  to  the 
plane  of  projection,  will  fall  in  the  parts  of  the  vertical 
lines  through  R  and  R,  which  lie  between  the  ridge  RR 
and  the  upper  line  of  the  front. 

27.  In  transferring  the  several  heights  from  the  section 
to  the  elevations,  each  dimension  was  measured  separately 


OF    OBLIQUE    ELEVATIONS. 


93 


R 

R 

r//            \ 

d 

"d 

- 

one  after  the  other;  but  it  is  best  to  transfer  the  various 
heights  from  a  section  to  an  elevation  all  at  once,  in  the 
same  manner  as  the  dimensions  are  transferred  from  the 
plan  to  the  elevation. 

Remember  that  the  section  of  the  cottage  was  formed  by 
a  plane  cutting  it  through  the 
door,  and  perpendicular  to  the 
front,  and  that  ar  is  the  line 
in  which  such  plane  cuts  the 
front.  At  a  convenient  dis- 
tance from  the  section  draw 
the  vertical  line  AR.  Then, 
through  the  various  points  of 
the  section  whose  heights  you 

wish  to  note,  draw  the  dotted  horizontal  lines  RR,  rr,  dd, 
and  Aa,  and  note  the  points  in  which  they  cut  the  vertical 
line  RA. 

The  distance  between  the  two  points  a  and  d  in  the  sec 
tion,  represents  the  height  of  the  door  in  the  cottage  ;  and 
the  distance  between  the  two  corresponding  points  A  and  d, 
on  the  vertical  line  AR,  will  represent  the  height  which 
ought  to  be  given  to  the  door  in  a  geometrical  elevation ; 
and  the  same  for  all  other  points.  Hence,  all  the  points 
necessary  for  transferring  the  several  heights  of  the  front 
of  the  building  from  a  section  to  an  elevation,  are  now 
marked  on  the  vertical  line  AR. 


28.  If  you  wish  to  draw  the  front  elevation  from  the  sec- 
tion and  plan,  draw  a  line  to  represent  the  ground  line. 
Then  draw  a  vertical  line  AR,  and  make  it  equal  to  ar, 
which  shows  the  height  of  the  cottage ;  and  lay  off  in  the 
same  manner  the  height  of  the  door  ad,  and  the  height  of  the 
wall  ar.  Then,  at  a  convenient  distance  from  a,  mark  the 
corner  of  the  cottage  1  ;  and  from  the  plan  lay  off  the  dis- 


94 


BOOK    III. SECTION    III. 


tances  from  1  to  2,  1  to  3,  4,  5,  and  6  ;  and  through  2t  3, 

R 


A  a  1      2    3         4     5       6 

and  6  draw  perpendiculars,  which  being  met  by  parallels 
through  d,  r,  and  R,  will  determine  all  the  parts  of  the  cot- 
tage. The  two  heights  aw,  aw,  to  the  bottom  and  top  of  the 
window,  are  not  found  in  the  section,  but  are  taken  from  the 
oblique  elevation. 

29.  After  finishing  an  elevation,  or  other  geometrical 
drawing,  the  superfluous  or  dotted  lines  representing  planes 
of  projection,  scales  of  heights,  &c,  are  rubbed  out ;  except- 
ing only  those  imaginary  lines,  marked  in  the  plan,  which 
show  the  direction  according  to  which  the  sections  or  ob- 
lique elevations  accompanying  the  plan  may  have  been 
taken. 

Let  us,  therefore,  rub  out  the  superfluous  lines,  letters, 
&c,  in  the  figures  which  have  been  drawn,  leaving  only 
such  as  are  necessary  to  explain  the  connection  between 
the  plan  and  section,  and  between  the  plan  and  oblique 
elevation.     We  shall  then  have — 


Jn  ez 


Plan. 


Section. 


GENERAL    REMARKS. 


95 


I. 

/fy-~A 


Front  Elevation.  Oblique  Elevation. 

The  Plan,  Section,  and  Elevation,  completed  above,  are 
sufficient  to  give  a  full  insight  into  the  principles  of  plan 
drawing. 


GENERAL    REMARKS. 

30.  All  plans,  sections,  and  elevations  are  drawn  by  lay- 
ing down  a  certain  number  of  points  and  lines  truly,  on 
some  plane  surface,  according  to  geometrical  principles.  In 
drawing  some  objects,  it  may  be  necessary  to  lay  down  a 
great  number  of  points  and  lines  :  in  others,  only  a  few  ;  but 
whether  the  number  be  great  or  small,  each  individual  point 
or  line  must  be  drawn,  in  all  cases,  according  to  some  one 
or  other  of  the  foregoing  rules. 

Plans  which,  as  before  stated,  resemble  the  appearance 
of  any  object  viewed  from  a  height  directly  above  it,  do  not 
carry  a  very  just  notion  of  the  object  to  persons  ignorant 
of  the  principles  of  plan  drawing ;  because  opportunities  of 
looking  perpendicularly  or  directly  down  upon  objects  are 
not  common. 

Ground  plans,  or  foundation  plans  of  buildings  or  other 
works,  do  not  give  any  just  notion  of  the  appearance  of  the 
object  represented ;  because  when  a  building  is  finished,  it 
is  impossible,  from  any  point  of  view  whatever,  to  see  the 
various  walls  and  foundations,  in  the  manner  in  which 
they  must  be  represented  in  the  ground  plan,  the  whole  of 
these  parts  being  hidden  by  the  roof.     In  fact,  the  ground 


96  BOOK    III. SECTION    III. 

plan  of  any  finished  building  is,  properly  speaking,  a  hori- 
zontal section  through  the  various  walls — the  only  difference 
between  it  and  the  common  section  consisting  in  this,  that 
the  common  section  is  taken  vertically,  whereas  the  sec- 
tion which  exhibits  the  ground  plan  is  taken  horizontally. 

31.  In  plans,  sections,  and  elevations  of  lany  object,  when 
the  various  points  and  lines  have  been  laid  down  according 
to  the  rules  of  projection,  it  is  usual  afterwards  to  color  or 
shade  the  figure  in  order  to  make  a  finished  drawing. 

The  art  of  plan  drawing,  therefore,  comprehends  two  dis 
tinct  operations  :  first,  the  projection  of  the  lines  which  form 
the  representation  of  the  object ;  and  secondly,  the  shading 
or  coloring  of  it.  .  ^j 

In  colored  plans  and  sections,  masonry  is  generally  ri     1 
red;  wood  so  as  to  represent  its  own  natural  color 
of  a  sandy  color;  iron  of  a  dark  blue  ;  and  water  of 'fiMlfci. 
ish  blue. 

In  plans  not  colored,  masonry  is  generally  made  "&aVk, 
while  wood  and  other  substances  are  made  lighter. 

In  sections  not  colored,  different  substances  are  shaded 
darker  or  lighter,  according  to  the  fancy  of  the  draughtsman. 

In  plans  of  buildings,  the  doors  and  windows  are  left 
blank,  while  the  walls  are  either  colored  or  shaded.  And 
in  sections,  a  marked  distinction  of  color  or  shade  is  also 
made  between  the  solid  part  of  the  walls,  and  the  doors, 
windows,  or  other  apertures  which  may  be  represented. 

In  elevations  of  any  object,  whether  colored  or  not,  the 
various  parts  are  shaded  in  such  a  manner  as  to  resemble, 
a 8  much  as  possible,  the  outward  appearance  of  the  object. 


OF   ARCHITECTURE.  87 


BOOK  IV. 


SECTION  I 


OF    ARCHITECTURE. 


J.   What  is  Architecture?  • 

architecture  is  the  art  of  construction. 

,  Int$  how  many  branches  is  it  divided? 
into  >three  principal  parts  : — 
st.  Civil  architecture,  which  embraces  the  construction 
of  public  and  private  edifices. 

2d.  Naval  architecture,  which  embraces  the  construction 
of  vessels,  ports,  artificial  harbors,  &c. ;  and 

3d.  Military  architecture,  which  embraces  the  construc- 
tion of  forts,  redoubts,  and  all  military  defences.  We  shall 
speak  here  only  of  civil  architecture. 

3.  What  are  the  elements  of  architecture  ? 
They  are  the  mouldings. 

4.  What  are  mouldings? 

They  are  the  projecting  parts  which  serve  to  ornament 
architecture. 

5.  How  many  kinds  of  mouldings  are  there  ? 

Three  kinds :  those  bounded  by  planes ;  those  bounded 
by  curved  surfaces ;  and  those  bounded  by  both  plane  and 
curved  surfaces. 


98  BOOK    IV. SECTION    I. 

6.  What  are  the  principal  plane  mouldings  ? 

They  are  the  Fillet,  the  Drip,  and  the  Plate-band. 

7.  What  is  a  fillet  ? 

It  is  a  square  moulding  which  projects  over  a  distance 
equal  to  its  height. 

8.  What  is  a  drip  ? 

It  is  a  large  projecting  moulding,  hollowed  on  the  under 
side,  and  placed  in  cornices  to  protect  the  edifice  from  rain 

9.  What  is  a  plate-band? 

It  is  a  large  and  flat  moulding  which  projects  but  little. 

10.  What  are  the- principal  circular  mouldings  ? 

The  Ovolo,  the  bead  or  Astragal,  the  Torus,  the  Cavetto 
the  Scotia,  the  Cyma-recta,  the  Cyma-reversa,  and  the  Ogee. 

11.  What  is  an   Ovolo,  and  how  do  you  trace  it? 

An  ovolo  is  a  moulding  flat  on  the  top  and  bottom,  and 
whose  circular  projection  is  equal  to  its  height. 

To  describe  it,  make  the  perpendicular  height  AD  equal 
to  the  projection  AC :  then,  with  A  as  a  centre,  describe 
the  arc  DC.  If  you  wish  to  make  a  flattened  ovolo,  with  B 
as  a  centre  and  BA  as  radius,  describe  an  arc  :  then,  with 
A  as  a  centre  and  AB  as  a  radius,  describe  a  second  arc, 
meeting  the  first  in  C.  Then,  with  C  as  a  ce  ntre,  describe 
the  arc  BA. 

12.  What  is  the  Bead  or  Astragal,  and  how  is  it  traced? 
It  is  a  thin  moulding,  of  which  the  circular  projection  is 

equal  to  half  the  height.     To  trace  it,  describe  L  semi-cir 
cumference,  of  which  the  diameter  AB  represents  the  height 
of  the  moulding. 

13.  What  is  a   Torus,  and  how  traced? 

It  is  a  moulding  similar  to  the  bead,  but  thicker.     It  is 


OF    ARCHITECTURE.  99 

traced  by  describing  a    semi-circumference  on  the  height 
AB  as  a  diameter. 

1 4.  What  is  a   Cavetto,  and  how  is  it  traced  ? 

A  cavetto  is  an  ovolo,  of  which  the  centre  C  is  in  a  per- 
pendicular from  the  extreme  projection  of  the  moulding.  It 
is  traced  by  describing  the  quarter  of  a  circumference  from  C 
as  a  centre.     The  second  figure  presents  a  cavetto  reversed. 

15.  What  is  a  Scotia,  and  how  is  it  traced  ? 

It  is  a  hollow  moulding,  formed  by  several  cavettos  with 
different  centres.  The  second  figure  represents  a  reversed 
scotia.  The  circular  parts  are  described  with  the  centres 
A  and  B. 

16.  What  is  a  Cyma-recta,  and  how  is  it  traced? 

The  cyma-recta  is  composed  of  an  ovolo  and  a  cavetto. 
To  describe  it,  draw  the  line  AB,  and  then  divide  the  pro- 
jection of  the  moulding  into  two  equal  parts  by  the  perpen- 
dicular CD,  and  produce  the  line  B :  the  point  D  will  be 
the  centre  of  the  ovolo,  and  the  point  C  of  the  cavetto, 
which  together  form  the  cyma-recta. 

The  flattened  cyma-recta  is  a  similar  moulding.  To 
trace  it,  it  is  necessary,  after  having  divided  the  line  AB 
into  two  equal  parts,  to  construct  an  equilateral  triangle  on 
each  of  the  parts.  The  points  C  and  D  will  then  be  the 
centres  of  the  arcs  which  form  the  moulding. 

17.  What  is  the  Ogee,  and  how  is  it  traced? 

The  ogee  is  a  moulding  composed  of  the  same  parts  as  the 
talon,  but  differently  placed.  Having  joined  the  points  A  and 
B,  we  draw  through  the  middle  point  of  this  line  the  line 
CD,  parallel  to  the  fillets  A  and  B,  and  the  points  C  and  D} 
m  which  it  meets  the  perpendiculars,  are  the  centres  of  the 
arcs  which  form  the  moulding.  If  the  ogee  is  flattened,  the 
centres  are  the  vertices  A  and  B  of  the  equilateral  triangles, 
each  constructed  on  the  half  of  DC. 


100  BOOK    IV. — SECTION    I. 

18*-.  How  do  you  trace  this  moulding,  when  its  proj  ction 
exceeds  its  height? 

Having  joined  the  points  A  and  B,  divide  it  into  two 
equal  parts  AC,  BC,  and  then  draw  IP  perpendicular  to 
CA  at  the  middle  point.  Next,  draw  LN  perpendicular  to 
B  at  the  middle  point,  but  in  a  contrary  direction.  Then 
draw  BN  perpendicular  to  the  fillet ;  after  which  draw  NC, 
and  produce  it  to  P :  then  P  and  N  will  be  the  centres  of 
the  arcs.  To  give  grace  to  this  moulding,  the  part  BC  is 
sometimes  made  shorter  than  the  part  CA :  in  every  other 
respect  the  construction  is  the  same. 

19.  How  are  these  mouldings  to  be  used  in  combination? 

They  are  not  to  be  used  at  hazard,  each  having  a  par- 
ticular situation  to  which  it  is  adapted,  and  where  it  must 
always  be  placed.  Thus,  the  ovolo  and  talon,  from  their 
peculiar  form,  seem  designed  to  support  other  important 
mouldings  ;  the  cyma  and  cavetto,  being  of  weaker  form, 
should  only  be  used  for  the  cover  or  shelter  of  the  other 
parts.  The  torus  and  astragal,  bearing  a  resemblance  to 
a  rope,  appear  calculated  to  bind  and  fortify  the  parts  to 
which  they  are  applied  ;  while  the  use  of  the  fillet  and  scotia 
is  to  separate  one  moulding  from  another,  and  to  give  a 
variety  to  the  general  appearance. 

The  ovolo  and  cyma  are  mostly  placed  in  situations  above 
the  level  of  the  eye  :  when  placed  below  it,  they  should  only 
be  applied  to  crowning  members.  The  place  of  the  scotia 
is  universally  below  the  level  of  the  eye.  When  the  fillet 
is  very  wide,  and  used  under  the  cyma  of  a  cornice,  it  is 
called  a  corona ;  if  under  a  corona,  it  is  called  a  band. 

The  curved  contours  of  mouldings  are  portions  either  of 
circles  or  ellipses :  the  Greeks  always  preferred  the  latter 


Fillet 


Plate-band 


Drip 


'    C 
Ovolo 


Bead 


Scotia 


Scotia 


Torus 


Cavetto 


D\  y'JA 
Cyma-recta 

T C 


&7JA 


Cyma-recta 


Tb"-'c 


Ovolo 


3=; 


j° 


Ovolo 


Cavetto-reversed 


Ci 


\A        D 


Cyma-reversa 


C       B\ 


Ogee  or  Talon 


'^*A 


Ogee 


IB 


'D 


ID 


T£ 


102  BOOK    IV  — SECTION    II. 


SECTION  II. 

OF    THE    ORDERS    OF     ARCHITECTURE,    AND    THEIR    PRINCIPAL 

PARTS. 

1.  How  many  orders  of  architecture  are  there? 

Five :  the  Tuscan,  the  Doric,  the  Ionic,  the  Corinthian, 
and  the  Composite. 

2.  How  many  parts  do  we  distinguish  in  each  of  the  five 
orders  ? 

Three  :  the  pedestal,  the  column,  and  the  entablature. 

3.  Of  how  many  parts  is  the  pedestal  composed? 
Three :  the  plinth,  the  die,  and  the  cornice. 

4.  Of  how  many  parts  is  the  column  composed? 
Three  :  the  base,  the  shaft,  and  the  capital. 

5.  Of  how  many  parts  is  the  entablature  composed? 
Three :  the  architrave,  the  frieze,  and  the  cornice. 

6.  Are  these  three  principal  parts  always  found  in  each  of 
the  orders? 

Not  always ;  for,  in  giving  the  name  of  an  order  to  an 
edifice,  regard  is  not  always  had  to  the  columns,  but  some- 
times to  the  proportions  observed  in  its  construction.  Some- 
times, even,  there  are  no  columns  ;  and  often  the  pedestal 
is  replaced  by  the  plinth  only. 

7.  How  are  the  five  orders  distinguished  ? 

The  Tuscan  is  distinguished  by  the  simplicity  of  its  mem- 
bers, having  no  ornament ;  the  Doric  by  the  triglyphs  which 
ornament  its  frieze ;  the  Ionic  by  the  volutes  of  its  capital ; 
the  Corinthian  by  the  leaves  which  ornament  its  capital ; 


Balustrade 


Volute 


fcZ£\L\kiA 


104  BOOK    IV. SECTION    II. 

and  the  Composite  by  the  Corinthian  capital,  united  with  the 
volutes  of  the  Ionic. 

8.  What  proportion  exists  between  the  diameter  and  height 
of  the  columns  in  the  different  orders  ? 

In  the  Tuscan  order,  the  height  of  the  column,  including 
its  base  and  capital,  is  seven  times  the  diameter  of  the  shaft 
at  the  base  ;  that  of  the  Doric  column  eight  times ;  that  of 
the  Ionic  nine  times ;  and  that  of  the  Corinthian  and  Com- 
posite ten  times. 

9.  What  proportions  are  established  between  the  three  prin- 
cipal parts,  in  the  orders  of  architecture  ? 

In  all  the  orders,  the  pedestal  is  one  third  the  height  of 
the  column,  and  the  entablature  is  one  quarter  the  height. 

10.  What  is  a  module? 

In  all  the  orders  except  the  Doric,  it  is  the  diameter  of 
the  shaft  at  the  base :  in  the  Doric,  it  is  the  semi-diameter. 
This  is  according  to  Gwilt's  Architecture,  published  at 
London  in  1839.  Some  authors  call  the  semi-diameter  a 
module. 

11.  What  is  a  minute? 

In  the  Doric  order,  the  module  is  supposed  to  be  divided 
into  thirty  equal  parts,  and  in  each  of  the  other  orders  into 
sixty,  and  each  of  the  equal  parts  is  called  a  minute :  hence 
a  minute  is  one -sixtieth  part  of  the  diameter  of  the  shaft 
at  its  base. 

12.  If  the  diameter  of  a  shaft  is  two  feet  at  the  base,  what 
will  be  the  height  of  the  structure  in  each  of  the  five  orders  ? 

FOR    THE    TUSCAN    ORDER. 

2  X  7  =  14  feet    =  height  of  column,  (Art.  8) 
add  one-third    =    4ft.  8in.  =  height  of  pedestal,  (Art.  9) 
add  one-fourth  =    3ft.  6in.  ==  height  of  entablature,  (Art.  9.) 


Total  height      =  22ft.  2in. 


Tuscan  Order. 


Doric  Order. 


106  BOOK    IV. SECTION    II. 

By  a  similar  process,  we  should  find  the  height  of  the 
Doric  to  be  25  feet  4  inches ;  that  of  the  Ionic,  28  feet 
6  inches  ;  that  of  the  Corinthian,  31  feet  8  inches ;  and  that 
of  the  Composite,  31  feet  8  inches. 

13.  What  is  the  form  of  the  shafts  of  the  columns? 
The  shafts  diminish  in  diameter  as  they  rise :  sometimes 

the  tapering  begins  at  the  foot  of  the  shaft ;  sometimes  from 
a  point  one  quarter  from  the  base,  and  sometimes  from  a 
point  one-third ;  and  in  some  examples  there  is  a  swelling 
in  the  middle.  The  difference  between  the  diameter  at  top 
and  that  at  bottom,  is  seldom  more  than  one-sixth  of  the 
least  diameter,  or  less  than  one -eighth. 

14.  What  do  you  remark  of  the  entablatures? 

The  entablature  and  its  subdivisions,  though  architects 
frequently  vary  from  the  proportions,  may  as  a  general  rule 
be  set  down  as  exhibited  in  the  drawings.  The  total  height 
of  the  entablature,  in  all  the  orders  except  the  Doric,  is 
divided  into  10  parts,  three  of  which  are  given  to  the  archi- 
trave, three  to  the  frieze,  and  four  to  the  cornice.  But  in 
the  Doric  order,  the  whole  should  be  divided  into  eight 
parts,  two  given  to  the  architrave,  three  to  the  frieze,  and 
three  to  the  cornice.  The  mouldings,  which  form  the  de- 
tail, of  these  leading  features,  are  best  learned  by  reference 
to  representations  of  the  orders  at  large. 

In  the  Ionic  order  the  entablatures  are  generally  very 
simple  The  architrave  has  one  or  two  fasciae ;  the  frieze 
is  plain,  and  the  cornice  has  four  parts.  In  the  Composite 
order,  the  entablature  is  large  for  so  slender  an  order ;  yet 
it  is  on  many  accounts  very  beautiful. 


Ionic  Order. 


Corinthian  Order. 


? 

e  i 

- f 

6  ! 

1 

-a   i 

? 

S5T 

j  ! 

1' 

e  i 

I 

r5  ! 

*-M    i   ■ 

1 

?   "\ 


MENSURATION    OF    SURFACES. 


109 


BOOK  V. 


SECTION   I. 

MENSURATION    OF    SURFACES. 

1.   What  do  you  understand  by  the  unit  of  length? 

If  the  length  of  a  line  be  computed  in  feet,  one  foot  is 
the  unit  of  the  line,  and  is  called  the  linear  unit. 

If  the  length  of  a  line  be  computed  in  yards,  one  yard 
is  the  linear  unit.  If  it  be  computed  in  rods,  one  rod  is 
the  linear  unit ;  and  if  it  be  computed  in  chains,  one  chain 
is  the  linear  unit. 


2.   What  do  you  understand  by  the  unit  of  surface  ? 


If  we  describe  a  square  on  the  unit 
of  length,  such  square  is  called  the  unit 
of  surface.  Thus,  if  the  linear  unit  be 
1  foot,  one  square  foot  will  be  the  unit 
of  surface. 


3.  How  many  square  feet  are  there  in 
a  square  yard? 

If  the  linear  unit  is  1  yard,  one  square 
yard  will  be  the  unit  of  surface  ;  and  this 
square  yard  contains  9  square  feet. 


1  yard  =  3  feet. 

no 


BOOK    V. SECTION    I. 


1  chain  =  4  rods. 


4.  How  many  square  rods  are  there 
in  a  square  chain  ? 

If  the  linear  unit  is  1  chain,  the 
unit  of  surface  will  be  1  square  chain, 
which  will  contain  16  square  rods. 


5.  How   do  you  find  the  number  of  square  feet  contained 
in  a  rectangle? 

If  we  have  a  rectangle  whose  base 
is  4  feet,  and  altitude  3  feet,  it  is  evi- 
dent that  it  will  contain  12  square 
feet.  These  12  square  feet  are  the 
measure  of  the  surface  of  the  rectan- 
gle. 


6.  How  do  you  find  the  number  of  squares  contained  in 
any  rectangle  ? 

It  is  plain  that  the  number  of  squares  in  any  rectangle, 
will  be  expressed  by  the  units  of  its  base,  multiplied  by  the 
units  in  its  altitude.  This  product  is  called  the  measure 
of  the  rectangle 

7.  What  do  you  mean  by  the  rectangle  of  two  lines  ? 

In  geometry,  we  often  say,  the  rectangle  of  two  lines, 
by  which  we  mean,  the  rectangle  of  which  those  lines  are 
the  two  adjacent  sides. 

8.  What  is  the  area  of  a  figure  ? 
The  measure  of  its  surface. 


9".   What  is  the  unit  of  the  number  which   expresses  the 
area? 

It  is  a  square,  of  which  the  linear  unit  is  the  side. 


MENSURATION    OF    SURFACES 


111 


10.  How  do  you  find  the  area  of  a  rectangle  ? 

The  area  of  a  rectangle  is  equal  to  the  product  of  its 
base  by  its,  altitude.  If  the  base  of  a  rectangle  is  30  yards, 
and  the  altitude  5  yards,  the  area  will  be  150  square  yards. 

11.  What  is  the  area  of  a  square  equal  to? 

The  area  of  a  square  is  equal  to  the  product  of  its  two 
equal  sides ;  that  is,  to  the  square  of  one  of  its  sides. 

12.  How  does  the  diagonal  of  a  rectangle  divide  it? 
The  diagonal  DB  divides  the  rect-        D Q 

angle  ABCD  into  two  equal  triangles. 
Hence,  a  triangle  is  half  a  rectangle, 
having  the  same  base  and  altitude. 

13.  What  is  the  altitude  -of  a  parallelogram? 
The  altitude  of  a  parallelogram  is 

the  perpendicular  distance  between 
two  of  its  parallel  sides.  Thus,  EB 
is  the  altitude  of  the  parallelogram 
ABCD. 


1 4.    What  part  of  a  parallelogram  is  a  triangle,  having  the 
same  base  and  the  same  altitude  ? 

A  triangle  is  also  half  a  parallelo- 
gram, having  the  same  base  and  alti- 
tude. 


15.  What  is  the  area  of  a  triangle 
The  area  of  a  triangle  is  equal  to 
half  the  product  of  the  base  by  the 
altitude ;  for,  the  base  multiplied  by 
the  altitude  gives  a  rectangle  which 
is  double  the  triangle.  Thus,  the  area 
of  the  triangle  ABC  is  equal  to  half 
the  product  of  AB  X  CD. 


iqual  to 


112  BOOK    V. SECTION    1. 

If. the  base  of  a  triangle  is  12,  and  the  altitude  8  yards 
the  area  will  be  48  square  yards. 

16.   What  is  the  area  of  a  parallelogram? 

The  area  of  a  parallelogram  is 
equal  to  its  base  multiplied  by  its 
altitude.  Thus,  the  area  of  the  par- 
allelogram  ABCD   is    equal   to  AB 


X  BE.  A 

If  the  base  is  20,  and  altitude  15  feet,  the  area  will  bo 
300  square  feet. 

17.   What  is  the  area  of  a  trapezoid  ? 

The   area  of  a  trapezoid  is  equal  to  jy          q 

half  the  sum  of  its  parallel  sides  multi-  j 

plied  by  the  perpendicular  distance  be-       Z 

tween  them.     Thus, 

area  ABCD  =  \{AB  +  CD)  x  CF. 


F  B 


18.  With  what  is  land  generally  measured? 
Surveyors,  in   measuring   land,    generally  use   a  chain, 

called  Gunter's  chain.     This  chain  is  four  rods,  or  66  feet 
in  length,  and  is  divided  into  100  links. 

19.  What  is  an  acre? 

An  acre  is  a  surface  equal  in  extent  to  10  square  chains , 
that  is,  equal  to  a  rectangle  of  which  one  side  is  ten  chains, 
and  the  other  side  one  chain. 

20.  What  is  a  quarter  of  an  acre  called  ? 
One  quarter  of  an  acre  is  called  a  rood. 

21.  How  many  square  rods  in  an  acre? 

Since  the  chain  is  4  rods  in  length,  1  square  chain  con- 
tains 16  square  rods;  and  therefore,  an  acre,  which  is  10 
square  chains,  contains  160  square  rods,  and  a  rood  con- 
tains 40  square  rods      The  square  rods  are  called  perches 


MENSURATION  OF  SURFACES.  113 

22.  How  is  land  generally  computed? 

Land  is  generally  computed  in  acres,  roods,  and  perches, 
which  are  respectively  designated  by  the  letters  A.  R.  P. 

23.  If  the  linear  dimensions  are  chains  or  links,  hew  do 
you  find  the  acres  ? 

When  the  linear  dimensions  of  a  survey  are  chains  or 
links,  the  area  will  be  expressed  in  square  chains  or  square 
links,  and  it  is  necessary  to  form  a  rule  for  reducing  this 
area  to  acres,  roods,  and  perches.  For  this  purpose,  let 
us  form  the  following 

TABLE. 

1  square  chain  =  10000  square  links. 
1  acre  =  10  square  chains  ==  100000  square  links. 

1  acre  =  4  roods  =160  perches. 
1  square  mile  =  6400  square  chains  =  640  acres. 

24-  If  the  linear  dimensions  are  links,  how  do  you  find  the 
acres  ? 

When  the  linear  dimensions  are  links,  the  area  will  be 
expressed  in  square  links,  and  may  be  reduced  to  acres  by 
dividing  by  100000,  the  number  of  square  links  in  an  acre  : 
that  is,  by  pointing  off  five  decimal  places  from  the  right 
hand. 

If  the  decimal  part  be  then  multiplied  by  4,  and  five 
places  of  decimals  pointed  off  from  the  right  hand,  the  fig- 
ures to  the  left  will  express  the  roods. 

If  the  decimal  part  of  this  result  be  now  multiplied  by 
40,  and  five  places  for  decimals  pointed  off,  as  before,  the 
figures  to  the  left  will  express  the  perches. 

If  one  of  the  dimensions  be  in  links,  and  the  other  in 
chains,  the  chains  may  be  reduced  to  links  by  annexing 
two  ciphers :  or,  the  multiplication  may  be  made  without 
annexing  the  ciphers,  and  the  product  reduced  to  acres  and 


114  BOOK    V. SECTION    I. 

decimals  of  an  acre,  by  pointing  off  three  decimal  places 
at  the  right  hand. 

When  both  the  dimensions  are  in  chains,  the  product  is 
reduced  to  acres  by  dividing  by  10,  or  pointing  off  one  deci 
mal  place. 

From  which  we  conclude  that, 

1st.  If  links  be  multiplied  by  links,  the  product  is  reduced 
to  acres  by  pointing  off  five  decimal  places  from  the  right 
hand. 

2d.  If  chains  be  multiplied  by  links,  the  product  is  re- 
duced to  acres  by  pointing  off  three  decimal  places  from 
the  right  hand. 

3d.  If  chains  be  multiplied  by  chains,  the  product  is  re- 
duced to  acres  by  pointing  off  one  decimal  place  from  the 
right  hand. 

25.  How  do  you  find  the  number  of  square  feet  in  an 
acre  ? 

Since  there  are  16.5  feet  in  a  rod,  a  square  rod  is 
equal  to 

16.5  X  16.5  =  272.25  square  feet. 

If  the  last  number  be  multiplied  by  160,  the  number  of 
square  rods  in  an  acre,  we  shall  have 

272.25  x  160  =  43560  ==  the  square  feet  in  an  acre. 

OF    THE    TRIANGLE. 

26.  How  do  you  find  the  area  of  a  triangle,  when  the  base 
and  altitude  are  known  ? 

1st.  Multiply  the  base  by  the  altitude,  and  half  the  product 
will  be  the  area. 

Or,  2d.  Multiply  the  base  by  half  the  altitude,  and  the 
product  will  be  the  area. 


MENSURATION    OF    SURFACES. 


115 


EXAMPLES. 


1.  Required  the  area  of  the  triangle 
ABC,  whose  base  AB  is  10.75  feet  and 
altitude  7.25  feet. 


We  first  multiply  the  base 
by  the  altitude,  and  then  di- 
vide the  product  by  2. 


Operation. 

10.75  X  7.25  =  77.9375 
and 
77.9375  ~2  =  38.96875 
==  area. 

2.  What  is  the  area  of  a  triangle  whose  base  is  18  feet 
4  inches,  and  altitude   11   feet  10  inches? 

Ans.   108  sq.ft   5/  8". 

3.  What  is  the   area  of  a  triangle  whose  base  is  12.25 
chains,  and  altitude  8.5  chains  ? 

Ans.  5  A.  0  R.  33  P. 

4.  What   is  the   area  of  a   triangle   whose  base    is  20 
feet,  and  altitude  10.25  feet? 

Ans.   102.5  sq.  ft. 

5.  Find  the  area  of  a  triangle  whose  base  is  625  and 
altitude  520  feet 

Ans.   162500  sq.ft. 

6.  Find  the  number  of  square  yards  in  a  triangle  whose 
base  is  40  and  altitude  30  feet. 

Ans.  66|  sq.  yds. 
7..  What  is  the   area  of  a  triangle  whose  base  is  72.7 
yards,  and  altitude  36.5  yards? 

Ans.  1326.775  sq.  yds. 

8.  What  is  the  content  of  a  triangular  field  whose  base 
is  25.01   chains,  and  perpendicular  18.14  chains? 

Ans.  22  A.  2  R.  29  P. 


116  BOOK    V. SECTION    I. 

9.  What  is  the  content  of  a  triangular  field  whoso  base 
is  15.48  chains,  and  altitude  9.67  chains  ? 

Ans.  7  A.  1  R.  38  P. 

27.  How  do  you  find  the  area  of  a  triangle  when  the  three 
sides  are  given  ? 

1st.  Add  the  three  sides  together  and  take  half  their  sum. 

2d.  From  this  half  sum  take  each  side  separately. 

3d.  Multiply  together  the  half  sum  and  each  of  the  three 
remainders,  and  then  extract  the  square  root  of  the  product, 
which  will  be  the  required  area. 

EXAMPLES. 

1.  Find  the  'area  of  a  triangle  whose  sides  are  20,  30, 
and  40  rods. 

20  45  45  45 

30  20 30       40 

40  25  1st  rem.      15  2d  rem.        5  3d  rem. 


2)90 
45  half  sum. 

Then,  to  obtain  the  product,  we  have 

45  x  25  X  15  X  5  =  84375; 
from  which  we  find 

area  =  ^84375  =  290.4737  perches. 

2.  How  many  square  yards  of  plastering  are  there  in  a 
triangle,  whose  sides  are  30,  40,  and  50  feet? 

Ans.  66|. 

3.  The  sides  of  a  triangular  field  are  49  chains,  50:25 
chains,  and  25.69  chains:  what  is  its  area? 

Ans.  61  A.  1  R.  39.68  P. 

4.  What  is  the  area  of  an  isosceles  triangle,  whose  base 
is  20,  and  each  of  the  equal  sides  15  ? 

Ans.  111.803. 


MENSURATION    OF    SURFACES.  117 

5.  How  many  acres  are  there  in  a  triangle  whose  three 
sides  are  380,  420,  and  765  yards? 

Ans.  9  A.  0  R.  38  P. 

6.  How  many  square  yards  in  a  triangle  whose  sides 
are   13,  14,  and  15  feet? 

Ans.  §\. 

7.  What  is  the  area  of  an  equilateral  triangle  whose  side 
is  25  feet? 

Ans.  270.6329  sq.ft. 

8.  What  is  the  area  of  a  triangle  whose  sides  are  24, 
36,  and  48  yards? 

Ans.  418.282  sq.  yds. 

28.  How  do  you  find  the  hypothenuse  of  a  right-angled 
triangle  when  the  base  and  perpendicular  are  known  ? 

1st.  Square  each  of  the  sides  separately. 

2d.  Add  the  squares  together. 

3d.  Extract  the  square  root  of  the  sum,  which  will  be 
the  hypothenuse  of  the  triangle. 

EXAMPLES. 

1.  In  the  right-angled  triangle  ABC,  we 
have 

AB  =  30  feet,  BC  =  40  feet, 
to  find  AC. 

We  first  square  each  side, 
and  then  take  the  sum,  of 
which  we  extract  the  square 
root,  which  gives 

AC  =  V2500  =  50  feet. 

2.  The  wall  of  a  building,  on  the  brink  of  a  river,  is  120 
feet  high,  and  the  breadth  of  the  river  70  yards :  what  is 


2500 


118 


BOOR    V. SECTION 


the  length  of  a  line  which  would  reach  from  the  top  of  the 
wall  to  the  opposite  edge  of  the  river? 

Ans.  241.86  ft. 

3.  The  side  roofs  of  a  house  of  which  the  eaves  are  of 
the  same  height,  form  a  right  angle  at  the  top.  Now,  the 
length  of  the  rafters  on  one  side  is  10  feet,  and  on  the  other 
14  feet :  what  is  the  breadth  of  the  house  ? 

Ans.  17.204  ft. 

4.  What  would  be  the  width  of  the  house,  in  the  last 
example,  if  the  rafters  on  each  side  were  10  feet? 

Ans.  14.142/*. 

5.  What  would  be  the  width,  if  the  rafters  on  each  side 
were  14  feet? 

Ans.  19.7989/*. 

29.  If  the  hypothenuse  and  one  side  of  a  right-angled  tri- 
angle are  known,  how  do  you  find  the  other  side  1 

Square  the  hypothenuse  and  also  the  other  given  side, 
and  take  their  difference :  extract  the  square  root  of  this 
difference,  and  the  result  will  be  the  required  side. 


EXAMPLES. 

1.  In   the   right-angled   triangle  ABC, 
there  are  given 

AC  =  50  feet,  and  AB  =  40  feet ; 
required  the  side  BC. 

We  first  square  the  hypo- 
thenuse and  the  other  side, 
after  which  we  take  the  dif- 
ference, and  then  extract  the 
square  root,  which  gives 

BC  z=  x/900  =  30  feet 


Operation. 

502  =  2500 

40*=:  1600 

Diff.  =    900 


AREA    OF    THE    SQUARE.  119 

2  The  height  of  a  precipice  on  the  brink  of  a  river  is 
103  feet,  and  a  line  of  320  feet  in  length  will  just  reach 
from  the  top  of  it  to  the  opposite  bank :  required  the  breadth 
of  the  river. 

Aiis.  302.9703 /*. 

3.  The  hypothenuse  of  a  triangle  is  53  yards,  and  the 
perpendicular  45  yards  :  what  is  the  base  ? 

Ans.  28  yds. 

4.  A  ladder  60  feet  in  length,  will  reach  to  a  window 
40  feet  from  the  ground  on  one  side  of  the  street,  and  by 
turning  it  over  to  the  other  side,  it  will  reach  a  window  50 
feet  from  the  ground :   required  the  breadth  of  the  street. 

Ans.  77.8875  ft. 

AREA  OF  THE  SQUARE. 

30.  How  do  you  find  the  area  of  a  square,  a  rectangle,  or 
a  parallelogram  ? 

Multiply  the  base  by  the  perpendicular  height,  and  the 
product  will  be  the  area. 

P C 

1.     Required   the*  area    of    the    square 
ABCD,  each  of  whose  sides  is  36  feet. 


Operation. 
36  x  36=  1296,??./*. 


We  multiply  two  sides  of 
the  square  together,  and  the 
product  is  the  area  in  square 
feet. 

2.  How   many   acres,   roods,   and  perches,  in   a   square 
whose  side-  is  35.25  chains  ? 

Ans.  124  A.  1  R.   1  P. 

3.  What  is  the  area  of  a  square  whose  side  is  8  feet 
4  inches?     (See  Arithmetic,  §  171.) 

Ans.  69  ft.  5'  4". 


120 


BOOK    V. SECTION    I. 


4.  What  is  the  content  of  a   square  field  whose  side  is 

46  rods  ? 

Ans.  13  A.  0  R.  36  P. 

5.  What  is  the    area  of  a  square  whose  side  is  4769 

yards  ? 

Ans.  22743361  sq.  yds. 


AREA    OF    THE    RECTANGLE. 


1.  To  find  the  area  of  a  rectangle 
AB  CD,  of  which  the  base  AB  =  45 
yards,  and  the  altitude  AD  =  15  yards. 


D 


Here  we  simply  multiply 
the  base  by  the  altitude,  and 
the  product  is  the  area. 


Operation. 
45  X  15  =  675  sq.  yds. 


2.  What  is  the   area   of  a  rectangle  whose  base  is  14 
feet  6  inches,  and  breadth  4  feet  9  inches? 

Ans.  68  sq.ft.  10/  6". 

3.  Find  the  area  of  a  rectangular  board  whose  length  is 

112  feet,  and  breadth  9  inches. 

Ans.  84  sq.ft. 

4.  Required  the  area  of  a  rectangle  whose  base  is  10.51, 

and  breadth  4.28  chains. 

Ans.  A  A.  1  R.  39.7  P  +. 

5.  Required  the  area  of  a  rectangle  whose  base  is  12 
feet  6  inches,  and  altitude  9  feet  3  inches. 

Ans.   115  sq.ft.  7' 6". 


AREA    OF    THE    PARALLELOGRAM. 


1.  What  is  the  area  of  the  paral- 
lelogram ABCD,  of  which  the  base 
AB  is  64  feet,  and  altitude  DE,  36 
feet? 


AREA    OF    THE    TilAPEZOIO. 


121 


Operation. 
64  X  36  =  2304  sq.ft. 


We  multiply  the  base  64, 
by  the  perpendicular  height 
36,  and  the  product  is  the 
required  area. 

2.  What  is  the  area  of  a  parallelogram  whose  base  is 

12.25  yards,  and  altitude  8.5  yards? 

Ans.  104.125  sq.  yds. 

3.  What  is  the  area  of  a  parallelogram  whose  base  is 
8.75  chains,  and  altitude  6  clhains? 

Ans.  5  A.  1  R.  0  P. 

4.  What  is  the  area  of  a  parallelogram  whose  base  is 
7  feet  9  inches,  and  altitude  3  feet  6  inches? 

Ans.  27  sq.ft.  V  6". 

5.  What  is  the  area  of  a  parallelogram  whose  base  is 
10.50  chains,  and  breadth  14.28  chains? 

Ans.  14  A.  3  R.  30  P+. 

AREA    OF    THE    TRAPEZOID. 

31.  How  do  you  find  the  area  of  a  trapezoid? 

Multiply  the  sum  of  the  parallel  sides  by  the  perpendicu- 
lar distance  between  them,  and  then  divide  the  product  by 
two :  the  quotient  will  be  the  area. 


EXAMPLES. 


D 


1.  Required  the  area  of  the  trapezoid         / 
ABCD,  having  given  L 

AB  =  321.51  ft.,  DC  =  214.24  ft.,  and  CE  =  171.16  ft. 

Operation. 


We  first  find  the  sum  of 
the  sides,  and  then  multi- 
ply it  by  the  perpendicular 
height,  after  which,  we  di- 
vide the  product  by  2,  for 
the  area. 


321.51+214.24=535.75  = 
sum  of  parallel  sides. 

Then, 
535.75x171.16  =  91698.97 
91698.97 


and, 

=  the  area. 


=  45849.485 


122  BOOK    V. SECTION    I. 

2.  What  is  the  area  of  %  trapezoid,  the  parallel  sides  of 
which  are  12.41  and  8.22  chains,  and  the  perpendicular 
distance  between  them  5.15  chains? 

Ans.  5  A.  1  R.  9.956  P. 

3.  Required  the  area  of  a  trapezoid  whose  parallel  sides 
are  25  feet  6  inches,  and  18  feet  9  inches,  and  the  per- 
pendicular distance  between  them  10  feet  and  5  inches. 

Ans.  230  sq.ft.  5'  7". 

4.  Required  the  area  of  a  trapezoid  whose  parallel  sides 
are  20.5  and  12.25,  and  the  perpendicular  distance  between 
them  10.75  yards. 

Ans.   176.03125  sq.  yds. 

5.  What  is  the  area  of  a  trapezoid  whose  parallel  sides 
are  7.50  chains,  and  12.25  chains,  and  the  perpendiculai 
height  15.40  chains? 

Ans.  15  A.  0  R.  32.2  P. 

6.  What  is  the  content  when  the  parallel  sides  are  20 
and  32  chains,  and  the  perpendicular  distance  between  them 
26  chains? 

Ans.  67  A.  2  R.  16  P. 

AREA    OF    A    QUADRILATERAL. 

32.  How  do  you  find  the  area  of  a  quadrilateral  ? 

Measure  the  four  sides  of  a  quadrilateral,  and  also  one 
of  the  diagonals  :  the  quadrilateral  will  thus  be  divided  into 
two  triangles,  in  both  of  which  all  the  sides  will  be  known. 
Then,  find  the  areas  of  the  triangles  separately,  and  their 
sum  will  be  the  area  of  the  quadrilateral. 

EXAMPLES. 

1.  Suppose  that  we  havte  meas-  y^*\. 

ured  the  sides  and  diagonal  AC,  //          \  ,   ^v 

of  the  quadrilateral  ABCD,  and  ^^\~        3p]       / 

found  ^\     |   / 


AREA    OF    A    QUADRILATERAL.  123 

AB  =  40.05  ch,    CD  =  29.87  ch, 
BC  ==  26.27  ch,   AD  =  37,07  ch, 

and  AC  =  55  ch: 

required  the  area  of  the  quadrilateral. 

Ans,  101  A.  1  R.  15  P. 

Remark. — Instead  of  measuring  the  four  sides  of  the 
quadrilateral,  we  may  let  fall  the  perpendiculars  Bb,  Dg% 
on  the  diagonal  A  C.  The  area  of  the  triangles  may  then 
be  determined  by  measuring  these  perpendiculars  and  the 
diagonal  AC.  The  perpendiculars  are  Dg  =  18.95  ch,  and 
Bb  =  17.92  ch. 

2.  Required  the  area  of  a  quadrilateral  whose  diagonal 
is  80.5  and  two  perpendiculars  24.5  and  30.1  feet. 

Ans.  2197.65  sq.jt. 

3.  What  is  the  area  of  a  quadrilateral  whose  diagonal  is 

108  feet  6  inches,  and  the  perpendiculars  56  feet  3  inches, 

and  60  feet  9  inches  1 

Ans.  6347  sq.ft.  3'. 

4.  How  many  square  yards  of  paving  in  a  quadrilateral 
whose  diagonal  is  65  feet,  and  the  two  perpendiculars  28 
and  33£  feet? 

Ans.  222^  sq.  yds. 

5.  Required  the  area  of  a  quadrilateral  whose  diagona> 
is  42  feet,  and  the  two  perpendiculars  18  and  16  feet. 

Ans.  714  sq.ft. 

6.  What   is   the    area  of  a  quadrilateral   in   which   the 

diagonal  is  320.75  chains,  and  the  two  perpendiculars  69.73 

chains,  and  130.27  chains? 

Ans.  3207  A.  2  R. 

33.  How  do  you  find  the  area  of  a  long  and  irregular  fig 
ure,  bounded  on  one  side  by  a  straight  line  ? 

1st.  Divide  the  right  line  or  base  into  any  number  of 


124  BOOK    V. SECTION    I. 

equal  parts,  and  measure  the  breadth  of  the  figure  at  the 
points  of  division,  and  also  at  the  extremities  of  the  baso. 

2d.  Add  together  the  intermediate  breadths,  and  half  the 
sura  of  the  extreme  ones. 

3d.  Multiply  this  sum  by  the  base  line,  and  divide  the 
product  by  the  number  of  equal  parts  of  the  base. 

EXAMPLES. 

1.  The  breadths  of  an  irregular  ^ 

figure,    at   five    equidistant   places,     % — k — -"'P~~    I        *& 
A,  B,  C,  D,  and  E,  being  8.20  ch,     i         I        I        I        I 
7.40  ch,    9.20  ch,    10.20  ch,   and     A      B      C      D      E 
8.60  chains,  and  the  whole  length  40  chains ;  required  the 
area. 

8.20  35.20 

8.60  40 

2)16.80  4)1408.00 

8.40  mean  of  the  extremes.      352.00  square  ch 
7.40 
9.20 
10.20 


35.20  sum. 

Ans.  35  A.  32  P. 

2.  The  length  of  an  irregular  piece  of  land  being  21  ch, 
and  the  breadths,  at  six  equidistant  points,  being  4.35  ch, 
5.15  ch,  3.55  ch,  4.12  ch,  5.02  ch,  and  6.10  chains  :  re- 
quired the  area.  Ans.  9  A.  2  R.  30  P. 

3.  The  length  of  an  irregular  figure  is  84  yards,  and  the 
breadths  at  six  equidistant  places  are  17.4;  20.6;  14.2; 
16.5;  20.1,  and  24.4:  what  is  the  area? 

Ans.   1550.64  sq.  yds. 

4.  The  length  of  an  irregular  field  is  39  rods,  and  its 
breadths  at  five  equidistant  places  are  4.8 ;  5.2  ;  4.1  ;  7.3, 
and  7.2  rods :  what  is  its  area  ? 

Ans.  220.35  sq.  rods. 


OF    POLYGONS. 


125 


5.  The  length  of  an  irregular  field  is  50  yards,  and  its 
breadths  at  seven  equidistant  points  are  5.5 ;  6.2 ;  7.3 ;  6 ; 
7.5 ;  7 ;  and  8.8  yards  :  what  is  its  area  ? 

Ans.  342.916  sq.  yds. 

6.  The  length  of  an  irregular  figure  being  37.6,  and  the 
breadths  at  nine  equidistant  places,  0  ;  4.4 ;  6.5  ;  7.6  ;  5.4  ; 
8 ;  5.2  ;  6.5 ;  and  6.1 :  what  is  the  area  ? 

Ans.  219.255 


OF    POLYGONS. 

34.   What  is  a  regular  Polygon  1 
A  regular  polygon  is   one  which 
has  all  its  sides  equal  to  each  other, 
each  to  each,  and  all  its  angles  equal 
to  each  other,  each  to  each. 

Thus,  if  the  polygon  ABCDE  be 
regular,  we  have 

AB  =  BC=CD  =  DE  =  EA  s 
angle  A  =  B  =  C  =  D  =  E. 


also 


35.   What  are  similar  polygons  1 

Similar  polygons  are  those  which  have  the  angles  of 
the  one  equal  to  the  angles  of  the  other,  each  to  each, 
and  the  sides  about  the 
equal  angles  propor- 
tional. Hence,  similar 
polygons  are  alike  in 
shape,  but  may  differ 
in  size. 

The  sides  which  are 
like    situated    in    two 
similar  polygons,  are    called  homologous   sides,  and   these 
sides  are  proportional  to  each  other. 


126 


BOOK    V. — SECTION    I. 


Thus,  if  ABODE 
and  FGHIK  are  two 
similar  polygons :  then 
angle  A  =  F,  B  =  G, 
C  =  H,  D  =  I,  and 
E  =  K. 


Also,  AB  :  FG 

and  AB  :  FG 

also,  AB  :  FG 

and  AB  :  FG 


BC  :   GH 
CD  :  HI; 
DE  :  IK 
EA  :  KF. 


36.  Into  how  many  triangles  may  any  polygon  be  divided  ? 

Any  polygon  may  be  divided  by  di- 
agonals, into  as  many  triangles  less 
two,  as  the  polygon  has  sides.  Thus, 
if  the  polygon  has  five  sides,  there  will 
be  three  triangles ;  if  it  has  six  sides, 
there  will  be  four ;  if  seven  sides,  five ; 
if  eight  sides,  six  ;  &c. 

37.  What  is  the  sum  of  all  the  in- 
ward angles  of  a  polygon  equal  to  ? 

The  sum  of  all  the  inward  angles 
of  any  polygon  is  equal  to  twice  as 
many  right  angles,  wanting  four,  as 
the  figure  has  sides.  Thus,  if  the 
polygon  has  five  sides,  we  have 

A  +  B  +  C  -\-D  +  E=\0  right  angles  —  4  right  angles 
=  6  right  angles. 

38.  What   is    the    sum    of  the    angles   of  a  quadrilateral 
equal  to? 

If  the  polygon  is   a  quadrilateral,  then  the  sum  of  the 

angles  will  be  equal  to  four  right  angles. 


OF    POLYGONS.  127 

39.  How  do  you  find  one  of  the  angles  of  a  regular  poly- 
gon? 

When  the  polygon  is  regular,  its  angles  will  be  equal  to 
each  other.  If,  then,  the  sum  of  the  inward  angles  be  di- 
vided by  the  number  of  angles,  the  quotient  will  be  the  value 
of  one  of  the  angles.  We  shall  find  the  value  in  degrees, 
by  simply  placing  90°  for  the  right  angle 

40.  How  do  you  find  one  of  the  angles  of  an  equilateral 
triangle  ? 

The  sum  of  all  the  angles  of  an  equilateral  triangle  is 
equal  to 

6  X  90°  -  4  X  90°  =  540°  —  360°  =  180° 
and  for  each  angle 

180°  —  3  =  60°: 
Hence,  each  angle  of  an  equilateral  triangle  is  equal  to 
60  degrees. 

41.  How  do  you  find  one  of  the  angles  of  a  square  or 
rectangle  ? 

The  sum  of  all  the  angles  of  a  square  or  rectangle  is 
8  X  90°  —  4  X  90°  =  720°  —  360°  =  360°  • 
and  for  each  of  the  angles 

360°  -r-  4  =  90°. 

42.  How  do  you  find  one  of  the  angles  of  a  regular  pen- 
tagon ? 

The  sum  of  all  the  angles  of  a  regular  pentagon  is 
equal  to 

10  X  90°  —  4  X  90°  =  900°  —  360°  =  540° : 
and  for  each  angle 

540°  -f-  5  =  108°. 

43.  How  do  you  find  one  of  the  angles  of  a  regular  hexa- 
gon? 

The  sum  of  all  the  angles  of  a  regular  hexagon  is  equal  to 


128 


BOOK    V. SECTION    I. 


12  X  90°  —  4  x  90a  =  1080°  —  360°  i=  720° : 
and  for  each  angle 

720°  -^  6  =  120°. 

44.  How  do  you  find  one  of  the  angles  of  a  regular  hepta- 
gon? 

The  sura  of  the  angles  of  a  regular  heptagon  is  equal  to 
14  x  90°  —  4  X  90°  =  1260°  -   360°  ==  900° : 
and  for  one  of  the  angles 

900°  h-  7=  128°  34' -f. 

45.  How  do  you  find  one  of  the  angles  of  a  regular  octOr- 
gon  ? 

The  sum  of  the  angles  of  a  regular  octagon  is  equal  to 
16  x  90°  —  4  X  90°  =  1440°  —  360°  =  1080°  • 
and  for  each  angle 

1080°  v8=  135°. 


46.  How  many  figures  can  be  arranged  about  a  point  so  as 
to  fill  up  the  entire  space  ? 

There  are  but  three ;  the  equilateral  triangle,  the  square 
or  rectangle,  and  the  hexagon. 

First. — Six  equilateral  triangles  placed 
about  the  point  C,  will  fill  the  entire 
space.  For,  each  angle  is  equal  to  60°, 
and  their  sum  to 

60°  x  6  =  360°. 


Second. — Four  squares,  or  rectangles, 
placed  about  Cf  will  fill  the  entire  space. 
For,  each  angle  is  equal  to  90°,  and 
their  sum  to 

90°  x  4  =  360°. 


£ 


OF    POLYGONS. 


129 


Third.  —  Three  hexagons 
placed  about  C,  will  fill  up 
the  entire  space.  For,  each 
angle  is  equal  to  120°,  and 
the  sum  of  the  three  to 
120°  x  3  =  360°. 


47.  How  are  similar  polygons  to  each  other? 

Similar  polygons  are 
to  each  other  as  the 
squares  described  on 
their  homologous  sides. 

Thus,  the  two  simi- 
lar polygons  ABCDE y 
FGHIK,  are  to  each 
other  as  the  squares 
described  on  the  homo- 
logous sides  AB  and 
FG :  that  is 

ABCDE  :  FGHIK  :  :  square  L  :  square  M. 

If  AB  were  4,  the  area  ABCDE  would  be  27.5276384. 
Now,  if  FG  were  8,  what  would  be  the  area  FGHIK? 


82 


27.5276384  :   110.1105536. 


48.  How  do  you  find  the  area  of  a  regular  polygon  1 
Multiply  half  the  perimeter  of  the  figure  by  the  perpen- 
dicular let  fall  from  the  centre  on  one  of  the  sides,  and  the 
product  will  be  the  area. 


130 


BOOK    V. SECTION 


EXAMPLES. 


1.  Required  the  area  of  the  regu- 
lar pentagon  ABCDE,  each  of  whose 
sides  AB,  BC,  &c,  is  25  feet,  and 
the  perpendicular  OP,  17.2  feet. 


We  first  multiply  one  side 
by  the  number  of  sides  and 
divide  the  product  by  2  :  this 
gives  half  the  perimeter, 
which  we  multiply  by  the 
perpendicular  for  the  area. 


25  X  5 


Operation. 
=  62.5  =  half  the  per- 
Then, 


2 
imeter. 

62.5  X  17.2  =  1075  sq.ft.  = 
the  area. 


2.  The  side  of  a  regular  pentagon  is  20  yards,  and  the 
perpendicular  from  the  centre  on  one  of  the  sides  13.76382  : 
required  the  area. 

Ans.  688.191  sq.  yds. 

3.  The  side  of  a  regular  hexagon  is  14,  and  the  perpen- 
dicular from  the  centre  on  one  of  the  sides  12.1243556 : 
required  the  area. 

Ans.  509.2229352  sq.ft. 

4.  Required  the  area  of  a  regular  hexagon  whose  side 
is  14.6,  and  perpendicular  from  the  centre  12.64  feet. 

Ans.  553.632  sq.ft. 

5.  Required  the  area  of  a  heptagon  whose  side  is  19.38, 
and  perpendicular  20  feet. 

Ans.  1356.6  sq.ft. 

6.  Required  the  area  of  an  octagon  whose  side  is  9.941 
yards  and  perpendicular  12  yards. 

Ans.  *477.168  sq.  yds. 

49.  The  following  table  shows  the  areas  of  the  ten  regu- 


OF    POLYGONS. 


131 


lar  polygons  when  the  side  of  each  is  equal  to  1 .     It  also 
shows  the  length  of  the  radius  of  the  inscribed  circle. 


Number  of 
sides. 

Names. 

Areas. 

Radius  of  inscribed 
circle. 

3 

Triangle, 

0.4330127 

0.2886751 

4 

Square, 

1.0000000 

0.5000000 

5 

Pentagon, 

1.7204774 

0.6881910 

6 

Hexagon, 

2.5980762 

0.8660^54 

7 

Heptagon, 

3.6339124 

1.0382617 

8 

Octagon, 

4.8284271 

1.2071068 

9 

Nonagon, 

6.1818242 

1.3737387 

10 

Decagon, 

7.6942088 

1.5388418 

11 

Undecagon, 

9.3656404 

1.2028437 

12 

Duodecagon, 

11.1961524 

1.8660254 

50.  How  do  you  find  the  area  of  any  polygon  from  the 
above  table? 

Since  the  areas  of  similar  polygons  are  to  each  other 
as  the  squares  described  on  their  homologous  sides,  we 
have 

Is  :  tabular  area  :   :  any  side  squared  :  area. 
Hence,  to  find  the  area  of  a  regular  polygon. 

1st.  Square  the  side  of  the  polygon. 

2d.  Multiply  the  square  so  found,  by  the  tabular  area  set 
opposite  the  polygon  of  the  same  number  of  sides,  and  the 
product  will  be  the  required  area. 


EXAMPLES. 

1.  What  is  the  area  of  a  regular  hexagon  whose   side 
is- 20?  • 

20s  =  400       and  tabular  area  =  2.5980762. 
Hence, 

2.5980762  x  400  =  1039.23048  =  the  area. 

2.  What  is  the  area  of  a  pentagon  whose  side  is  25  ? 

Ans.   1075.298375. 


132  BOOK    V. SECTION    1. 

3.  What  is  the  area  of  a  heptagon  whose  side  is  30* 

Ans.  3270.52116. 

4.  What  is  the  area  of  an  octagon  whose  side  is  10  feet? 

Ans.  482.84271  sq.ft. 

5.  The  side  of  a  nonagon  is  50 :  what  is  its  area  ? 

Ans.  15454.5605. 
i 

6.  The  side  of  an  undecagon  is  20 :  what  is  its  area? 

Ans.  3746.25616. 

7.  The  side  of  a  duodecagon  is  40 :   what  is  its  area  ? 

Ans.  17913.84384. 

8.  Required  the  area  of  an  octagon  whose  side  is  16. 

Ans.  1236.0773. 

9.  Required  the  area  of  a  decagon  whose  side  is  20.5. 

Ans.  3233.4912. 

10.  Required  the  area  of  a  nonagon  whose  side  is  36. 

'  Ans.  8011.6442. 

11.  Required  the  area  of  a  duodecagon  whose  side  is  125 

Ans.  174939.881. 


OF    THE    CIRCLE. 

51.  How  are  the  circumferences  of  circles  to  each  other? 

The  circumferences 
of  circles  are  propor- 
tional to  their  diame- 
ters. If  we  represent 
the  diameter  AB  by  Z), 
and  the  circumference 
of  the  circle  by  C,  and  the  diameter  CD  by  d,  and  the  cii 
cumference  by  c,  we  shall  have 

D    :    d    :  :    C    :    c, 


OF    THE    CIRCLE.  133 

52.  How  many  times  greater  is  the  circumference  than  the 
diameter  of  a  aircle  ? 

The  circumference  of  a  circle  is  a  little  more  than  three 
times  greater  than  the  diameter.  If  the  diameter  is  1,  the 
circumference  will  be  3.1416. 

53.  What  is  the  area  of  a  circle  equal  to  1 
The  area  of  a  circle  is  equal  to  the 

product  of  half  the  radius,  into  the  cir- 
cumference.    Thus,  the  area  of  the  cir- 
cle whose  centre  is  C,  is  equal  to  half     B{ 
the  radius  CA,  multiplied  by  the  circum- . 
ference :  that  is, 

area  ==  ±CA  x  circumference  ABB. 

54.  How  do  you  find  the  circumference  of  a  circle  when  tha 
diameter  is  known  ? 

Multiply  the  diameter  by  3.1416,  and  the  product  will 
be  the  circumference. 

EXAMPLES. 

1.  What  is  the  circumference  of  a  circle  whose  diame^ 
ter  is  17? 

We    simply   multiply    the 


number  3.1416  by  the  diam- 
eter, and  the  product  is  the 
circumference. 


Operation. 

3.1416  x  17  =  53.4072 

which  is  the  circumference 


2.  What  is  the  circumference  of  a  circle  whose  diameter 
is  40  feet  1 

Ans.  125.664/* 

3.  What  is  the  circumference  of  a  circle  whose  diame- 
ter is   12  feet? 

Ans.  37.6992  ft,    ' 


134  BOOK    V. SECTION    I. 

4.  What  is  the  circumference  of  a  circle  whose  diame- 
ter is  22  yards? 

Ans.  69.1152  yds. 

5.  What  is  the  circumference  of  the  earth — the  mean 
diameter  being  about  7921  miles  ? 

Ans.  24884.6136  miles. 

55.  How  do  you  find  the  diameter  of  a  circle  when  the  cir 
cumference  is  known? 

Divide  the  circumference  by  the  number  3.1416,  and  the 
quotient  will  be  the  diameter. 


EXAMPLES. 

1.  The  circumference  of  a  circle  is  69.1152  yards .  what 
is  the  diameter  ? 

Operation. 


We  simply  divide  the  cir- 
cumference by  3.1416,  and 
the  quotient  22  is  the  diame- 
ter sought. 


3.1416)69.1152(22 
62832 
62832 
62832 


2.  What  is  the  diameter  of  a  circle.whose  circumference 
is  11652.1944  feet? 

Ans.  3709  ft. 

3.  What  is  the  diameter  of  a  circle  whose  circumference 

is  6850? 

Ans.  2180.4176. 

4.  What  is  the  diameter  of  a  circle  whose  circumference 
is  50? 

Ans.  15.915. 

5.  If  the  circumference  of  a  circle  is  25000.8528,  what 

is  the  diameter? 

Ans.  7958. 


OF    THE    CIRCLE.  135 

56.  How  do  you  find  the  length  of  a  circular  arc,  when  the 
number  of  degrees  which  it  contains,  and  the  radius  of  the 
circle  are  known  ? 

Multiply  the  number  of  degrees  by  the  decimal  .01745, 
and  the  product  arising  by  the  radius  of  the  circle. 


EXAMPLES. 

1.  What  is  the  length  of  an  arc  of  30  degrees,  in  a 
circle  whose  radius  is  9  feet? 


We  merely  multiply  the 
given  decimal  by  the  num- 
ber of  degrees,  and  by  the 
radius. 


Operation. 
.01745  X  30  X9  =  4.7115, 
which  is  the    length  of  the 
required  arc. 


Remark. — When  the  arc  contains  degrees  and  minutes, 
reduce  the  minutes  to  the  decimals  of  a  degree,  which  is 
done  by  dividing  them  by  60. 

2.  What  is  the  length  of  an  arc  containing  12°  10'  or 
12£°,  the  diameter  of  the  circle  being  20  yards  ? 

Ans.  2.1231. 

3.  What  is  the  length  of  an  arc  of  10°  15'  or  101°,  in 
a  circle  whose  diameter  is  68? 

Ans.  6.0813. 

57.  How  do  you  find  the  length  of  the  arc  of  a  circle  when 
the  chord  and  radius  are  given  ? 

1st.  Find  the  chord  of  half  the  arc. 

2d.  From  eight  times  the  chord  of  half  the  arc,  subtract 
the  chord  ot  the  whole  arc,  and  divide  the  remainder  by 
three,  and  the  quotient  will  be  the  length  of  the  arc, 
nearly 


136 


BOOK    V. — SECTION    I. 


EXAMPLES. 

1.  The  chord  AB  =  30  feet,  and 
the  radius  AC  =  20  feet:  what  is 
the  length  of  the  arc  ABB  ? 

First,  draw  CD  perpendicular  to 
the  chord  AB :  it  will  bisect  the 
chord  at  P,  and  the  arc  of  the  chord 
at  D.     Then  AP  =  15  feet.     Hence 


AC  -  AP   =CF:    that  is 

400  —  225  =  175,  and  Vl75=  13.228  =  CP, 
Then,     CD  —  CP  =20  -  13.228  =  6.772  =  DP. 


Again,      AD  =  ^~A?+  TT?  =  V  225  -f  45.859984  : 
hence,       AD  =  16.4578  =  chord  of  the  half  arc. 
16.4578  X  8  -  30 


Then, 


33.8874  =  arc  ADB. 


2.  What  is  the  length  of  an  arc,  the  chord  of  which  is 
24  feet,  and  the  radius  of  the  circle  20  feet  ? 

Ans.  25.7309  ft. 

3.  The  chord  of  an  arc  is  16,  and  the  diameter  of  the 
circle  20  :    what  is  the  length  of  the  arc  ? 

Ans.  18.5178. 

4.  The  chord  of  an  arc  is  50,  and  the   chord  of  half 
the  arc  is  27 :   what  is  the  length  of  the  arc  ? 

Ans.  551. 

58.  How  do  you  find  the  area  of  a  circle  when  the  diametet 
and  circumference  are  both  known? 

Multiply  the  circumference  by  half  the   iadius,  and  the 
product  will  be  the  area. 


OF    THE    CIRCLE.  137 

EXAMPLES. 

1.  What  is  the  area  of  a  circle  whose  diameter  is  10, 
and  circumference  31.416? 
If  the  diameter  be  10,  the 


radius  is  5,  and  half  the  ra- 
dius 2i :  hence  the  circumfe- 
rence multiplied  by  21  gives 
the  area. 


Operation. 
31.416  X  21  =  78.54, 
which  is  the  area. 


2.  Find  the  area  of  a  circle  whose  diameter  is  7,  and 
circumference  21.9912  yards. 

Ans.  38.4846  yds. 

3.  How  many  square  yards  in  a  circle  whose  diameter 
is  31  feet,  and  circumference  10.9956? 

Ans.  1.069016 

4.  What  is  the  area  of  a  circle  whose  diameter  is  100, 
and  circumference  314.16? 

Ans.  7854 

5.  What  is  the   area  of  a   circle  whose   diameter  is  1, 
and  circumference  3.1416  ? 

Ans.  0.7854. 

6.  What  is  the  area  of  a  circle  whose  diameter  is  40, 
and  circumference  131.9472? 

Ans.  1319.472. 

59.  How  do  you  find  the  area  of  a  circle  when  the  diam- 
eter only  is  known? 

Square  the  diameter,  and  then  multiply  by  the   decimal 
.7854. 

EXAMPLES. 

1.  What  is  the  area  of  a  circle  whose  diameter  is  5  ? 


138 


BOOK    V. SECTION    I. 


We  square  the  diameter, 
which  gives  us  25,  and  we 
then  multiply  this  number 
and  the  decimal  .7854  to- 
gether. 


Operation. 

.7854 
?=      25 


39270 
15708 


area  =  19.6350 


2    What  is  the  area  of  a  circle  whose  diameter  is  7? 

Ans.  38.4846. 

3.  What  is  the  area  of  a  circle  whose  diameter  is  4.5  % 

Ans.   15.90435. 

4.  What  is  the  number  of  square  yards  in  a  circle  whose 
diameter  is  1}  yards  ? 

Ans.  1.069016. 

5.  What  is  the  area  of  a  circle  whose  diameter  is  8  75 
feet? 

Ans.  60.1322  sq.ft. 

60.  How  do  you  find  the  area  of  a   circle  when  the  cir- 
cumference only  is  known? 

Multiply  the  square  of  the  circumference  by  the  decimal 
07958,  and  the  product  will  be  the  area  very  nearly. 


EXAMPLES. 

1.  What  is  the  area  of  a  circle  whose  circumference  is 
3.1416? 


We  first  square  the  cir- 
cumference, and  then  multi- 
ply by  the  decimal  .07958. 


Operation. 


3.1416 


9.86965056 
.07958 


area  =  .7854+ 


2.  What  is  the  area  of  a  circle  whose  circumference 

is  91? 

Ans.  659.00198. 


OF    THE    CIRCLE.  139 

3.  Suppose  a  wheel  turns  twice  in  tracking  16£  feet,  and 
that  it  turns  just  200  times  in  going  round  a  circular  bowl- 
ing-green :   what  is  the  area  in  acres,  roods,  and  perches  ? 

Ans.  4  A.  3  R.  35.8  P. 

4.  How  many  square  feet  are  there  in  a  circle  whose 
circumference  is  10.9956  yards? 

Ans.  86.5933. 

5.  How  many  perches  are  there  in  a  circle  whose  cir- 
cumference is  7  miles? 

Ans.  399300.608. 

61.  Having  given  a  circle,  how  do  you  find  a  square  which 
shall  have  an  equal  area? 

1st.  The  diameter  X  .8862  =  side  of  an  equivalent 
square. 

2d.  The  circumference  X  .2821  =  side  of  an  equivalent 
square. 

EXAMPLES. 

1.  The  diameter  of  a  circle  is  100:  what  is  the  side 
of  a  square  of  an  equal  area  ? 

Ans.  88.62. 

2.  The  diameter  of  a  circular  fish-pond  is  20  feet :  what 
would  be  the  side  of  a  square  fish-pond  of  an  equal  area  ? 

Ans.  17.724  ft. 

3.  A  man  has  a  circular  meadow,  of  which  the  diameter 
is  875  yards,  and  wishes  to  exchange  it  for  a  square  one 
of  equal  size  :   what  must  be  the  side  of  the  square  ? 

Ans.  775.425. 

4.  The  circumference  of  a  circle  is  200:  what  is  the 
side  of  a  square  of  an  equal  area  ? 

Ans   56.42. 


140  BOOK    V. SECTION    1 

5.  The  circumference  of  a  round  fish-pond  is  400  yards 
what  is  the  side  of  a  square  fish-pond  of  equal  area? 

Ans.  112.84. 

6.  The  circumference  of  a  circular  bowling-green  is  412 
yards :  what  is  the  side  of  a  square  one  of  equal  area  ? 

Ans.  116.2252  yds. 

62.  Having  given  the  diameter  or  circumference  of  a  circle, 
how  do  you  find  the  side  of  the  inscribed  square  ? 

1st.  The  diameter  x  .7071  =  side  of  the  inscribed 
square. 

2d.  The  circumference  X  .2251  ==  side  of  the  inscribed 
square. 

EXAMPLES. 

1.  The  diameter  AB  of  a  cir- 
cle is  400 :  what  is  the  value  of 
AC,  the  side  of  the  inscribed 
square  ? 

Here 
.7071  x  400  =  282.8400  =  AC. 

2.  The  diameter  of  a  circle  is  412  feet:  what  is  the 
side  of  the  inscribed  square  ? 

Ans.  291.3252/*. 

3.  If  the  diameter  of  a  circle  be  600,  what  is  the  side 
of  the  inscribed  square  ? 

Ans.  424.26. 

4.  The  circumference  of  a  circle  is  312  feet:  what  is 
the  side  of  the  inscribed  square  ? 

Ans.  70.2312/*. 

5.  The  circumference  of  a  circle  is  819  yards:  what  is 
the  side  of  the  inscribed  square  ? 

Ans.  184.3569  yds 


OF    THE    CIRCLE.  141 

6.  The  circumference  of  a   circle  is  715 :    what  is  tho 
side  of  the  inscribed  square  ? 

Ans.  160.9465. 

63.  How  do  you  find  the  area  of  a  circular  sector  ? 
1st.  Find  the  length  of  the  arc  by  Art.  56. 

2d.  Multiply  the  arc  by  one-half  the  radius,  and  the  pro- 
duct will  be  the  area. 

EXAMPLES. 

1.  What  is  the  area  of  the  circular 
sector  ACB,  the  arc  AB  containing  18°, 
and  the  radius  CA  being  equal  to  3  feet  1 

First,  .01745  x  18  X  3  =  .94230  == 
length  AB. 

Then,  .94230  xl}  =  1.41345  =  area. 

2.  What  is  the  area  of  a   sector  of  a  circle,  in  which 
the  radius  is  20  and  the  arc  one  of  22  degrees? 

Ans.  76.7800. 

3.  Required  the  area  of  a  sector  whose  radius  is  25  and 
the  arc  one  of  147°  29'. 

Ans.  804.2448. 

4.  Required  the  area  of  a  semicircle  in  which  the   ra- 
dius is  13. 

Ans.  265.4143. 

5.  What  is  the  area  of  a  circular  sector  when  the  length 
of  the  arc  is  650  feet  and  the  radius  325  1 

Ans.  105625  sq.ft. 

64.  How  do  you  find  the  area  of  a  segment  of  a  circle  ? 
1st.  Find  the  area  of  the  sector  having  the  same   arc 

with  the  segment,  by  the  last  problem. 

2d.  Find  the  area  of  the   triangle  formed  by  the   chord 
of  the  segment  and  the  two  radii  through  its  extremities. 


142 


BOOK    V.— SECTION    I. 


3d.  If  the  segment  is  greater  than  the  semicircle,  add 
the  two  areas  together ;  but  if  it  is  less,  subtract  them,  and 
the  result  in  either  case  will  be  the  area  required. 


EXAMPLES. 

1.  What  is  the  area  of  the  seg- 
ment ADB,  the  chord  AB  —  24  feet, 
and  CA  =  20  feet  ? 

First,       CP  =  VcT-AP 


—  ^400  —  144 
then, 
PD=zCD-  CP 


20 


and, 
then, 


Vap 


AD 
arc  ADB  = 


V144  +  16  =  12.64911: 
HI  X  8  -24 


*  +  PD2 


arc 

half  radius 
area  sector 
area 


ADB  —  25.7309 

= 10 

ADBC  =  257.309 
CAB  =  192 


25.7309 

AP  =    12 

CP  =    16 

area  CAB  ==  192 


65.309  =  area  of  segment  ADB. 


2.  Find  the  area  of  the  segment 
AFB,  knowing  the  following  lines, 
viz. :    AB  =  20.5  :     FP  =?  17.17  ; 


AF  =  20  ;  FG 
11.64. 


Arc   AGF  = 


11.5,  and  CA  =z 


FG  X8-AF 


3  3 

sector  AGFBC  =  24  x  11.64  ==  279.36  : 

but         CP  =  FP  —  AC  =  17.17  —  11.64  =  5.53  : 


Then,  area  ACB  = 


AB  x  CP       20.5x5.53 


=  56.6825. 


OF    THE    CIRCLE.  143 

Then,  area  of  sector  AFBC  =  279.36 

do.    of  triangle  ABC  =    56.6825 
gives  area  of  segment    AFB  =  336.0425 

3.  What  is  the  area  of  a  segment,  the  radius  of  the  cir- 
cle being  1.0,  and  the  chord  of  the  arc  12  yards? 

Ans.  16.324  sq.  yds. 

4.  Required  the  area  of  the  segment  of  a  circle  whose 
chord  is  16,  and  the  diameter  of  the  circle  20. 

Ans.  44.5903. 

5.  What  is  the  area  of  a  segment  whose  arc  is  a  quad- 
rant— the  diameter  of  the  circle  being  18? 

Ans.  63.6174. 

6.  The  diameter  of  a  circle  is  100,  and  the  chord  of  the 
segment  60 :  what  is  the  area  of  the  segment  ? 

Ans.  408,  nearly. 

65.  How  do  you  jind  the  area  of  a  circular  ring ;  that  is, 
the  area  included  between  the  circumferences  of  two  circles, 
having  a  common  centre? 

1st.  Square  the  diameter  of  each  ring,  and  subtract  the 
square  of  the  less  from  that  of  the  greater. 

2d.  Multiply  the  difference  of  the  squares  by  the  decimal 
,7854,  and  the  product  will  be  the  area. 

EXAMPLES. 

1.  In  the  concentric  circles 
having  the  common  centre  C, 
we  have 

AB  =  10  yards,  and  BE  — 
6  yards :    what  is  the  area  of 
the     space    included    between 
them? 


144  BOOK    V. SECTION    I. 


A&  = 

To2 

=  100 

DE%  = 

62 

=    36 

DifFere 

rice 

=    64 

64 

X 

.7854  = 

50.2656  = 

area. 

Then, 

2.  What  is  the  area  of  the  ring  when  the  diameters  of 
the  circles  are  20  and  10  ? 

Ans.  235.62. 

3.  If  the  diameters  are  20  and  15,  what  will  be  the  area 
included  between  the  circumferences? 

Ans.   137.445. 

4.  If  the  diameters  are  16  and  10,  what  will  be  the  area 
included  between  the  circumferences? 

Ans.  122.5224. 

5.  Two  diameters  are  21.75  and  9.5;  required  the  area 
of  the  circular  ring. 

Ans.  300.6609. 

6.  If  the  two  diameters  are  4  and  6,  what  is  the  area 
of  the  ring? 

Ans.   15.708 

66.  How  do  you  find  the  area  of  an  ellipse  ? 
Multiply  the  two  axes  together,  and  their  product  by  the 
decimal  .7854,  and  the  result  will  be  the  required  area. 


EXAMPLES. 

1.  Required  the  area  of  an  ellipse, 
whose  transverse  axis  AB  =  70  feet, 
and  the  conjugate  axis  DE  =  50 
feet. 

AB  x  DE  =  70  x  50  =  3500 : 
Then,  .7854  x  3500  =  2748.9  =  area. 


MENSURATION    OF    SOLIDS.  145 

2.  Required  the   area  of  an  ellipse  whose  axes  are  24 

and  18. 

Ans.  339.2928. 

3.  What  is  the  area  of  an  ellipse  whose  axes   are  35 

and  25  ? 

Ans.  687.225. 

4.  What  is  the   area  of  an   ellipse  whose   axes  are  80 

and  60  ? 

Ans.  3769.92. 

5.  What  is  the  area  of  an  ellipse  whose  axes  are  50 

and  45? 

Ans.  1767.15. 


SECTION   II. 


MENSURATION    OF    SOLIDS. 


1.  What  is  a  solid  or  body  ? 

A  solid  or  body  is  that  which  has  length,  breadth,  and 
thickness. 

2.  What  is  a  body  bounded  by  planes  called? 

Every  solid  bounded  by  planes  is  called  a  polyedron. 

3.  What  are  the  bounding  planes,  the  straight  lines,  and 
the  angular  points  called? 

The  planes  which  bound  a  polyedror  are  called  faces. 
The  straight  lines  in  which  the  faces  intersect  each  other, 
are  called  the  edges  of  the  polyedron;  and  the  points  at 
which  the  edges  intersect,  are  called  the  vertices  of  the 
angles,  or  vertices  of  the  polyedron. 

7 


146 


BOOK    V. SECTION    II. 


4.    What  is  a  prism  ?    What  are  its  bases  ?  what  its  con- 
vex surface  ? 

A  prism  is  a  solid,  whose  ends 
are  equal  polygons,  and  whose  side 
faces  are  parallelograms. 

Thus,  the  prism  whose  lower  base 
is  the  pentagon  AB CD E,  terminates 
in  an  equal  and  parallel  pentagon 
FGHIK,  which  is  called  the  upper 
base.  The  side  faces  of  the  prism 
are  the  parallelograms  DH,  DK,  EF, 
AG,  BH.  These  are  called  the 
convex  or  lateral  surface  of  the  prism. 


5.    What  is  the  altitude  of  a  prism? 

The  altitude  of  a  prism  is  the  distance  between  its  upper 
and  lower  bases ;  that  is,  it  is  a  line  drawn  from  a  point 
of  the  upper  base,  perpendicular  to  the  lower  base. 


6.   What  is  a  right  prism  ? 

A  right  prism  is  one  in  which 
the  edges  AF,  BG,  EK,  HC,  and 
DI  are  perpendicular  to  the  bases. 
In  the  right  prism,  either  of  the  per- 
pendicular edges  is  equal  to  the  al- 
titude. In  the  oblique  prism  the 
altitude  is  less  than  the  edge. 


7.  How  are  prisms  distinguished  from  each  other  ? 

A  prism  whose  base  is  a  triangle,  is  called  a  triangular 
prism  :  if  the  base  is  a  quadrangle,  it  is  called  a  quad- 
rangular prism  :  if  a  pentagon,  a  pentagonal  prism :  if  a 
hexagon,  it  is  called  a  hexagonal  prism:    &c. 


MENSURATION    OF    SOLIDS. 


147 


8.    What  is  a  parallelopipedon  ?    what  a  cube  ? 

A  prism  whose  base  is  a  parallelogram,  and  all  of  whose 
faces  are  also  parallelograms,  is  called  a 
parallelopipedon.  If  all  the  faces  are  rec- 
tangles, it  is  called  a  rectangular  paral- 
lelopipedon. If  all  the  faces  are  squares, 
it  is  called  a  cube.  The  cube  is  bounded 
by  six  equal  faces  at  right  angles  to  each 
other. 


\ 


9.  How  do  the  opposite  faces  of  a  parallelopipedon  com- 
pare  with  each  other?  jj 

The  opposite  faces  of  a  parallelopi- 
pedon are  equal  to  each  other.  Thus, 
the  parallelogram  BD  is  equal  to  the 
opposite  parallelogram  FH,  the  parallel- 
ogram BE  to  CH,  and  BG  to  AH. 


Dr~- 


10.   What  is  the  content  of  a  solid  ? 

The  content  of  a  solid  is  the  number  of  cubes  which  it 
contains. 

In  order  to  find  the  content  of  a  solid,  suppose  ABCD 
to  be  the  base  of  a  parallelo- 
pipedon. 

Let  us  suppose  AB  ==  4  feet, 
and  BC  =  3  feet.  Then  the 
number  of  square  feet  in  the 
base  will  be  equal  to  3x4  = 
12  square  feet.  Therefore,  12 
equal  cubes  of  one  foot  each, 
may  be  placed  by  the  side  of  each  other  on  the  base. 
If  the  parallelopipedon  be  1  foot  in  height,  it  will  contain 
12  such  cubes,  or  12  cubic  feet :  were  it  2  feet  in  height, 
it  would   contain   two   tiers    of  cubes,  or   24  cubic   feet: 


148  BOOK    V. SECTION    II. 

were  it  3  feet  in  height,  it  would  contain  three  tiers  of 
cubes,  or  36  cubic  feet.  Therefore,  the  solid  content  of  a 
parallelopipedon  is  equal  to  the  product  of  its  length,  breadth, 
and  height. 

11.  How  many  kinds  of  quantity  are  there  in  geometry? 

There  are  three  kinds  of  quantity  in  geometry,  viz.  : 
Lines,  Surfaces,  and  Solids ;  and  each  of  these  has  its 
own  unit. 

12.  What  are  the  units  of  these  kinds  of  quantity  ? 
The   unit  of  a  line,  which  we   have    called  the   linear 

unit,  is  a  line  of  a  known  length,  as  a  foot,  a  yard,  a 
rod,  &c. 

The  unit  of  surface  is  a  square,  whose  sides  are  the 
unit  of  length. 

The  unit  of  solidity  is  a  cube,  whose  edges  are  the  unit 
of  length. 

For  example,  if  the  bounding  lines  of  a  surface  be  es- 
timated in  yards,  the  content  will  be  square  yards  ;  and 
if  the  bounding  lines  of  a  solid  be  yards,  its  surface  will 
be  estimated  in  square  yards,  and  its  solid  content  in  cubic 
yards. 

13.  Into  how  many  parts  is  the  mensuration  of  solids  di- 
vided ? 

The  mensuration  of  solids  is.  divided  into  two  parts: — 
1st.  The  mensuration  of  the  surfaces  of  solids :    and 
2dly.  The  mensuration  of  their  solidities. 

14.  How  is  a  curved  line  to  be  treated? 

A  curve  line  which  is  expressed  by  numbers  is  also 
referred  to  a  unit  of  length,  and  its  numerical  value  is 
the  number  of  times  which  the  line  contains  the  unit. 

If,  then,  we  suppose  the  linear  unit  to  be  reduced  to  a 


OF    THE    PRISM. 


149 


straight  line,  and  a  square  constructed   on   this  line,  this 
square  will  be  the  unit  of  measure  for  curved  surfaces 

15.  Repeat  the  table  of  solid  measures. 

1  cubic  foot     =1728  cubic  inches. 
1  cubic  yard    =  27  cubic  feet. 
1  cubic  rod      =  4492|  cubic  feet. 
1  ale  gallon     =282  cubic  inches. 
1  wine  gallon  =231  cubic  inches. 
1  bushel  =  2150.42  cubic  inches. 

OF    THE    PRISM. 

16.  How  do  you  find  the  surface  of  a  right  prism? 
Multiply  the  perimeter  of  the  base  by  the  altitude,  and 

the  product  will  be  the  convex  surface  :    and  to  this   add 
the  area  of  the  bases  when  the  entire  surface  is  required. 


EXAMPLES. 

1.  Find  the  entire  surface  of  the 
regular  prism,  whose  base  is  the 
regular  polygon  ABCDE,  and  al- 
titude AF,  when  each  side  of  the 
base  is  20  feet,  and  the  altitude  AF 
50  feet.      . 

AB  +  BC  +  CD  +  DE  +  EA  = 
100 ;    and  AF  =  50  : 
then  (AB  +  BC  +  CD+DE  +  EA)X  AF  =  convex  surface 
becomes  100  X  50  =  5000  square  feet,  which  is  the  con- 
vex surface.     For  the  area  of  the  end,  we  have 
AB    x  tabular  number  =  area  ABCDE ,  (see  page  131 ;) 
that  is,     202  x  tabular  number,  or  400  X  1  720477  = 
688.1908  =lhe  area  ABCDE. 


150  BOOK    V. SECTION    II. 

Then    convex  surface  =  5000  square  feet, 

lower  base  688.1908         do. 

upper  base  688.1908         do. 

entire  surface         6376.3816         do. 

2.  What  is  the  surface  of  a  cube,  the  length  of  each 
side  being  20  feet-?  isJ,  2400  sq.ft. 

3.  Find  the  entire  surface  of  a  triangular  prism,  whose 
base  is  an  equilateral  triangle,  having  each  of  its  sides 
equal  to  18  inches,  and  altitude  20  feet. 

Ans.  91.949  sq.ft. 

4.  What  is  the  convex  surface  of  a  regular  octagonal 
prism,  the  side  of  whose  base  is  15  and  altitude  12  feet? 

Ans.  1440  sq.ft. 

5.  What  must  be  paid  for  lining  a  rectangular  cistern 
with  lead  at  2c?.  a  pound,  the  thickness  of  the  lead  being 
such  as  to  require  lib.  for  each  square  foot  of  surface : 
the  inner  dimensions  of  the  cistern  being  as  follows :  viz., 
the  length  3  feet  2  inches,  the  breadth  2  feet  8  inches,  and 
the  depth  2  feet  6  inches?  ^    £2  &   ^ 

17.  How  do  you  find  the  solidity  of  a  prism,  parallelopi- 
pedon,  or  cube? 

Multiply  the  area  of  the  base  by  the  perpendicular  height, 
and  the  product  will  be  the  solidity. 

EXAMPLES. 


1.  What  is  the  solidity  of  a  regular 
pentagonal  prism,  whose  altitude  is  20, 
and  each  side  of  the  base  15  feet? 

To  find  the  area  of  the  base  we 
have  (page  131) 


OF    THE    PRISM.  *  151 

152  =  225:  and  225  x  1.7204774  =  387.107415  =  the 
area  of  the  base  :    hence, 

387.107415  X  20  =  7742.1483  =  solidity. 

2.  What  is  the  solid  content  of  a  cube  whose  side  is  24 
mches  ?  Ans.  13824  solid  in. 

3.  How  many  cubic  feet  in  a  block  of  marble,  of  which 

the  length  is  3  feet  2  inches,  breadth  2  feet  8  inches,  and 

height  or  thickness  2  feet  6  inches? 

Ans.  211  solid  ft. 

4.  How  many  gallons  of  water,  ale  measure,  will  a  cis- 
tern contain  whose  dimensions  are  the  same  as  in  the  last 
example  ? 

Ans.  12911. 

5.  Required  the  solidity  of  a  triangular  prism,  whose  al- 
titude is  10  feet,  and  the  three  sides  of  its  triangular  base 
3,  4,  and  5  fee* 

Ans.  60  solid  ft. 

6.  What  is  the  solidity  of  a  square  prism,  whose  height 
is  5i  feet,  and  each  side  of  the  base  li  feet? 

Ans.  9%  solid  ft. 

7.  What  is  the  solidity  of  a  prism,  whose  base  is  an 
equilateral  triangle,  each  side  of  which  is  4  feet,  the  height 
of  the  prism  being  10  feet? 

Ans.  69.282  solid  ft. 

8.  What  is  the  number  of  cubic  or  solid  feet  in  a  reg- 
ular pentagonal  prism,  of  which  the  altitude  is  15  feet  and 
each  side  of  the  base  3.75  feet  ? 

Ans.  362.913. 

9.  What  is  the   solidity   of  a  prism,  whose  base  is  an 

equilateral  triangle,  each  side  of  which  is  1.5  feet,  and  the 

altitude  18  feet?  .        fWi.-A1ift.      ,.    a 

Ans.   17.53701435  cubic  ft. 


152 


BOOK    V. SECTION    II. 


10.  What  is  the  solidity  of  a  cube,  whose   side  is   15 

inches  ? 

Ans.  1.953125  cubic  ft. 

11.  What  is  the  solidity  of  a  cube,  whose  side  is  17.5 
inches  ? 

Ans.  3.1015  cubic  ft. 

12.  What  is  the  solidity  of  a  prism,  whose  base  is  a 
hexagon,  each  side  of  which  is  1  foot  4  inches,  and  the 
length  of  the  prism  15  feet? 

Ans.  69.2820285  solid  ft. 

13.  What  is  the  solidity  of  a  prism,  whose  altitude  is 
30  feet,  and  whose  base  is  a  heptagon,  each  side  of  which 
is  13  feet  3  inches? 

Ans.  1913.936237175  solid  ft. 


OF    THE    PYRAMID. 


18.  What  is  a  pyramid,  and  what  are  its  parts  ? 

A  pyramid   is    a   solid,  formed   by  # 

several  triangles  united  at  the  same 
point  S,  and  terminating  in  the  dif- 
ferent sides  of  a  polygon  ABCDE. 

The  polygon  ABCDE,  is  called  the 
base  of  the  pyramid ;  the  point  S  is 
called  the  vertex,  and  the  triangles 
ASB,  BSC,  CSD,  DSE,  and  ESA, 
form  its  lateral,  or  convex  surface. 

19.  What  is  a  solid  angle,  and  what  is  tlie  least  number 
of  planes  that  can  form  one  ? 

A  solid  angle  is   the   angular  spaces   included  between 
several   planes  which   meet   at  a  point.     Thus,  the   solid 


OF    THE     PYRAMID. 


153 


angle  S  is  formed  by  the  meeting  of  the  five  planes  ESD, 
DSC,  CSB,  BSA,  and  ASE.  The  point  S  is  called  the 
vertex  of  the  solid  angle.  Three  planes,  at  least,  are  re- 
quired to  form  a  solid  angle. 


20.  What  is  the  altitude  of  a  pyra- 
mid ? 

The  altitude  of  a  pyramid,  is  the 
perpendicular  let  fall  from  the  vertex 
upon  the  plane  of  the  base.  Thus, 
SO  is  the  altitude  of  the  pyramid  S 
-ABCDE. 


21.  What  is  the  slant  height  of  a 
pyramid  1 

The  slant  height  of  a  regular  pyra- 
mid, is  a  line  drawn  from  the  ver- 
tex, perpendicular  to  one  of  the  sides 
of  the  polygon  which  forms  its  base. 
Thus,  SF  is  the  slant  height  of  the 
pyramid  S— ABCDE. 


22.    What  is  the  axis  of  a  pyramid  1 

When  the  base  of  the  pyramid  is  a  regular  polygon,  and 
the  perpendicular  SO  passes  through  the  middle  point  of 
the  base,  the  pyramid  is  called  a  regular  pyramid,  and  the 
line  SO  is  called  the  axis. 

7* 


154 


BOOK    V. SECTION    II. 


23.  What  is  the  frustum  of  a  pyra- 
mid? what  the  altitude  of  the  frus- 
tum ? 

If  from  the  pyramid  S—ABCDE 
the  pyramid  S — abode  be  cut  off  by 
a  plane  parallel  to  the  base,  the  re- 
maining solid,  below  the  plane,  is 
called  the  frustum  of  a  pyramid. 

The  altitude  of  a  frustum  is  the 
perpendicular  distance  between  the 
upper  and  lower  planes. 

24.  How  are  pyramids  distinguished? 

A  pyramid  whose  base  is  a  triangle,  is  called  a  tri- 
angular pyramid ;  if  the  base  is  a  quadrangle,  it  is  called 
a  quadrangular  pyramid ;  if  a  pentagon,  it  is  called  a  pen- 
tagonal pyramid ;  if  the  base  is  a  hexagon,  it  is  called  a 
hexagonal  pyramid,  &c. 

25.  What  is  the  convex  surface  of 
a  regular  pyramid  equal  to  ? 

The  convex  surface  of  a  regular 
pyramid,  is  equal  to  the  perimeter  of 
the  base,  multiplied  by  half  the  slant 
height.  Thus,  the  convex  surface  of 
the  pyramid  S — ABCDE  is  equal  to 

$SF  (AB  +  BC+CD  +  DE  +  EA.) 


26.  How  do  you  find  the  surface  of  a  regular  pyramid  ? 

Multiply  the  perimeter  of  the  base  by  half  the  slant 
height,  and  the  product  will  be  the  convex  surface :  to 
this  add  the  area  of  the  base,  if  the  entire  surface  is  re- 
quired. 


OF    THE    PYRAMID. 


155 


EXAMPLES. 

1.  In  the  regular  pentagonal  pyra- 
mid   S—ABCDE,   the    slant   height 
SF  is  equal  to  45,  and  each  side  of 
the    base    is    15   feet :    required   the 
convex  surface,  and  also  the   entire 
surface. 
15  x  5=75=  perimeter  of  the  base, 
75  X  221  —  1687.5  square  feet  = 
area  of  convex  surface. 

•   And  15*  =  225, 
then  225  X  1.7204774  =  3S7.107415  = 


the  area  of  the  base. 


Hence,  convex  surface  =  1687.5 

area  of  the  base     =    387.107415 
entire  surface 


2074.607415  square  feet. 


2.  What  is  the  convex  surface  of  a  regular  triangular 
pyramid,  the  slant  height  being  20  feet,  and  each  side  of 
the  base  3  feet?  ^    90  sq.ft. 

3.  What  is  the  entire  surface  of  a  regular  pyramid, 
whose  slant  height  is  15  feet,  and  the  base  a  regular  pen 
tagon,  of  which  each  side  is  25  feet  1 

Ans.  2012.798  sq.ft. 

4.  What  is  the  entire  surface  of  a  regular  octagonal 
pyramid,  of  which  each  side  of  the  base  is  9.941  yards, 
and  the  slant  height  15  ?  ^    1073  628  ^  yd$ 

27.  How  do  you  find  the  solidity  of  a  pyramid  ? 
Multiply  the  area  of  the  base  by  the  altitude,  and  divide 
the  product  by  three :  the  quotient  will  be  the  solidity. 


156 


BOOK    V. SECTION    II. 


EXAMPLES. 

1.  What  is  the  solidity  of  a  pyra- 
mid, the  area  of  whose  base  is  215 
square  feet,  and  the  altitude  SO  =  45 
feet? 

First,       215  x  45  =  9675: 
then         9675  -j-  3  =  3225 

which  is  the  solidity  expressed  in  solid 
feet. 


2.  Required  the  solidity  of  a  square  pyramid,  each  side 
.of  its  base  being  30,  and  its  altitude  25. 

Ans.  7500  solid  ft. 

3.  How  many  solid  yards  are  there  in  a  triangular  pyra- 
mid, whose  altitude  is  90  feet,  and  each  side  of  its  base 

3yards?  ^n,.  38.97114. 

4.  How  many  solid  feet  in  a  triangular  pyramid,  the  al- 
titude of  which  is  14  feet  6  inches,  and  the  three  sides 
of  its  base  5,  6,  and  7  feet?  ^    71.0352. 

5.  What  is  the  solidity  of  a  regular  pentagonal  pyramid, 
its  altitude  being  12  feet,  and  each  side  of  its  base  2  feet? 

Ans.  27.5276  solid  ft. 

6.  How  many  solid  feet  in  a  regular  hexagonal  pyra- 
mid, whose  altitude  is  6.4  feet,  and  each  side  of  the  base 

6  inches?  ,         1  no^cA 

An?.  1.38564. 

7.  How  many  solid  feet  are  contained  in  a  hexagonal 
pyramid,  the  height  of  which  is  45  feet,  and  each  side 
of  the  base  10  feet?  ^  3897.1143. 


OF  THE  FRUSTUM  OF  A  PYRAMID.         157 

8.  The  spire  of  a  church  is  an  octagonal  pyramid,  each 
side  of  the  base  being  5  feet  10  inches,  and  its  perpen- 
dicular height  45  feet.  Within  is  a  cavity,  or  hollow  part, 
each  side  of  the  base  of  which  is  4  feet  11  inches,  and 
its  perpendicular  height  41  feet :  how  many  yards  of  stone 
does  the  spire  contain?  An^  33<197353> 


OF  THE  FRUSTUM  OF  A  PYRAMID. 

28.  How  do  you  find  the  convex  surface  of  the  frustum  of 
a  regular  pyramid  ? 

Multiply  half  the  sum  of  the  perimeters  of  the  two  bases 
by  the  slant  height  of  the  frustum,  and  the  product  will  be 
the  convex  surface. 

EXAMPLES. 

1.  In  the  frustum  of  the  regular  pen- 
tagonal pyramid  each  side  of  the  lower 
base  is  30  and  each  side  of  the  upper 
base  is  20  feet,  and  the  slant  height  fF 
is  equal  to  15  feet.  What  is  the  convex 
surface  of  the  frustum  1 

Ans.   1875  sq.ft. 

2.  How  many  square  feet  are  there  in  the  convex  sur- 
face of  the  frustum  of  a  square  pyramid,  whose  slant  height 
is  10  feet,  each  side  of  the  lower  base  3  feet  4  inches,  and 
each  side  of  the  upper  base  2  feet  2  inches  ? 

Ans.  110. 

3.  What  is  the  convex  surface  of  the  frustum  of  a  hep- 
tagonal  pyramid  whose  slant  height  is  55  feet,  each  side 
of  the  lower  base  8  feet,  and  each  side  of  the  upper  base 

4feet?  Ans.  2310*?  ft. 


158  BOOK    V. — SECTION    II. 

29.  How  do  you  find  the  entire  surface  of  the  frustum  of 
a  regular  pyramid? 

To  the   convex  surface,  found  as   above,  add  the  areas 
of  the  two  ends,  and  the  result  will  be  the  entire  surface. 
What  is  the  entire    surface  of  the    frustum  in  each  of 
the  last  three  examples? 

r  1st.  4111.62062  sq.ft. 
Ans.    )2d.  125|f  sq.ft. 

(3d.  2600.712992  sq.ft. 

30.  How  do  you  find  the  solidity  of  the  frustum  of  a 
pyramid  1 

Add  together  the  areas  of  the  two  bases  of  the  frustum 
and  a  geometrical  mean  proportional  between  them ;  and 
then  multiply  the  sum  by  the  altitude  and  take  one-third 
of  the  product  for  the  solidity. 


EXAMPLES. 

1.  What  is  the  solidity  of  the  frus- 
tum of  a  pentagonal  pyramid,  the  area 
of  the  lower  base  being  16  and  of  the 
upper  base  9  square  feet,  the  altitude 
being  7  feet? 


First,    16x9  =  144  :  then  -/l44  =  12  the  mean. 

Then,  area  of  lower  base  =  16 

"  upper  base  =r    9 

mean  of  bases    =  12 

37 

height  7 

3)259 
solidity     =  86J-  solid  feet. 


OF    THE    CYLINDER.  159 

2.  What  is  the  number  of  solid  feet  in  a  piece  of  tim- 
ber ^  nose  bases  are  squares,  each  side  of  the  lower  base 
bei  15  inches,  and  each  side  of  the  upper  base  being  6 
inc>      —the  length  being  24  feet? 

i  Ans.   19.5. 

6.  Required  the  solidity  of  a  regular  pentagonal  frus- 
tum, whose  altitude  is  5  feet,  each  side  of  the  lower  base 
18  inches,  and  each  side  of  the  upper  base  six  inches. 

Ans.  9.31925  solid  ft. 

4.  What  is  the  content  of  a  regular  hexagonal  frustum, 
whose  height  is  6  feet,  the  side  of  the  greater  end  18 
inches,  and  of  the  less  end  12  inches  1 

Ans.  24.681724  cubic  ft. 

5.  How  many  cubic  feet  in  a   square  piece  of  timber, 

the  areas  of  the  two  ends  being  504  and  372  inches,  and 

its  length  31i  feet? 

Ans.  95.447. 

6.  What  is  the  solidity  of  a  square  piece  of  timber,  its 
length  being  18  feet,  each  side  of  the  greater  base  18 
inches,   and  each  side  of  the  smaller  12  inches  ? 

Ans.  28.5  cubic  ft. 

7.  What  is  the  solidity  of  the  frustum  of  a  regular  hex- 
agonal pyramid,  the  side  of  the  greater  end  being  3  feet, 
that  of  the  less  2  feet,  and  the  height  12  feet? 

Ans.  197.453776  solid  ft. 


OF    THE    CYLINDER. 

31.  What  is  a  cylinder  ?  what  its  upper,  and  what  its 
lower  base?     What  is  the  axis? 

A  Cylinder  is  a  solid,  described  by  the  revolution  of  a 
rectangle  AEFD,  about  a  fixed  side  EF. 


160 


BOOK    V. SECTION    II. 


As  the  rectangle  AEFD  turns  aro? 
the  side  EF,  like  a  door  upon  its  hiir 
the   lines  AE  and  FD  describe  cir 
and  the    line   AD  describes    the    c 
surface  of  the  cylinder.  B 

The  circle  described  by  the  line  rp<    L  \,  j    ^ 

is  called  the  lower  base  of  the  cyli, 
and  the  circle  described  by  DF  is  c 
the  upper  base. 

The  immoveable  line  EF  is  called  the  ylin- 

der. 

A    cylinder,  therefore,   is    a    round   body   with    circular 
ends. 


32.  How  will  a  plane,  passed  through 
the  axis,  cut  the  cylinder? 

If  a  plane  be  passed  through  the  axis 
of  a  cylinder,  it  will  intersect  it  in  a 
rectangle  PG,  which  is  double  the  re- 
volving rectangle  EB. 


33.  If  a  cylinder  be  cut  by  a  plane 
parallel  to  the  base,  how  is  the  section  ? 

If  a  cylinder  be  cut  by  a  plane  paral- 
lel to  the  base,  the  section  will  be  a 
circle  equal  to  tl  e  base.  Thus,  MLKN 
is  a  circle  equal  to  the  base  FGC. 


OF    THE    CYLINDER 


161 


34.  How  do  you  find  the  surface  of  a  cylinder  ? 

The  convex  surface  of  a  cylinder  is 
equal  to  the  circumference  of  the  base, 
multiplied  by  the  altitude.  Thus,  the  con- 
vex surface  of  the  cylinder  A  C,  is  equal 
to  circumference  of  base  x  AD :  to  this 
add  the  areas  of  the  two  ends,  when  the 
entire  surface  is  required. 


EXAMPLES. 

1.  What  is  the  entire  surface  of  the 
cylinder,  in  which  AB,  the  diameter  of 
the  base,  is  12  feet,  and  the  altitude  EF 
30  feet  ? 

First,  to  find  the  circumference  of  the 
base,  (see  page  133,)  we  have 
3.1416  x  12  =  37.6992  ==  circumference 
of  the  base. 


Then, 

37.6992  X  30  =  1130.9760  =  convex 

surface. 

Also, 

122  = 

144  :  and  144  X 

.7854  =  113.0976  =  ar 

the  base. 

Then, 

convex  surface 
lower  base 
upper  base 
Entire  area 

=  1130.9760 
113.0976 
113.0976 

=  1357.1712 

2.  What  is  the  convex  surface  of  a  cylinder,  the  diam- 
eter of  whose  base  is. 20,  and  altitude  50  feet? 

Ans.  3141.6  s%.ft. 

3.  Required  the  entire  surface  of  a  cylinder,  whose  alti- 
tude is  20  feet,  and  the  diameter  of  the  base  2  feet. 

Ans.  1 31 .9472  ft. 


162 


BOOK    V.  — SECTION    II. 


4.  What  is  the  convex  surface  of  a  cylinder,  the  diam 
eter  of  whose  base  is  30  inches,  and  altitude  5  feet? 

Ans.   5654.88  sq.  inches. 

5.  Required  the  convex  surface  of  a  cylinder,  whose 
altitude  is  14  feet,  and  the  circumference  of  the  base  8 
feet  4  inches.  Ans.  116.6666,  &c,  sq.ft. 


35.  How  do  you  find  the  solidity  of  a 
cylinder  ? 

The  solidity  of  a  cylinder  is  equal  to 
the  area  of  the  base,  multiplied  by  the 
altitude.  Thus,  the  solidity  of  the  cylin- 
der AC  is  equal  to 

area  of  base  x  FE.  A 


EXAMPLES. 

1.  What  is  the  solidity  of  a  cylinder, 
the  diameter  of  whose  base  is  40  feet, 
and  altitude  EF,  25  feet? 

First,  to  find  the  area  of  the  base,  we 
have,  (see  page  137,) 

40s  ==  1600,  then  1600  X  .7854  = 
1256.64  =  area  of  the  base.     Then, 
1256.64  X  25  =  31416    solid    feet, 
which  is  the  solidity. 

2.  What  is  the  solidity  of  a   cylinder,  the  diameter   of 
whose  base  is  30  feet,  and  altitude  50  feet? 

Ans.  35343  cubic  ft. 

3.  What  is  the  solidity  of  a   cylinder  whose  height  is 
5  feet,  and  the  diameter  of  the  end  2  feet? 

Ans.   15.708  solid  ft. 

4.  What  is   the    solidity  of  a    cylinder  whose  height  is 
20  feet,  and  the  circumference  of  the  base  20  feet? 

Ans.  636.64  cubic  ft 


OF    THE    CONE.  163 

5.  The  circumference  of  the  base  of  a  cylinder  is  20 
feet,  and  the  altitude  19.318  feet:  what  is  the  solidity? 

Ans.  614.93  cubic  ft. 

6.  What  is  the  solidity  of  a  cylinder  whose  altitude  is 
12  feet,  and  the  diameter  of  its  base   15  feet? 

Ans.  2120.58  cubic  ft. 

7.  Required  the  solidity  of  a  cylinder  whose  altitude  is 
20  feet,  and  the  circumference  of  whose  base  is  5  feet 
6  inches.  Ans.  48.1459  cubic  ft. 

8.  What  is  the  solidity  of  a  cylinder,  the  circumference 
of  whose  base  is  38  feet,  and  altitude  25  feet  ? 

Ans.  2872.838  cubic  ft. 

9.  What  is  the  solidity  of  a  cylinder,  the  circumference 
of  whose  base  is  40  feet,  and  altitude  30  feet  ? 

Ans.  3819.84  solid  ft. 

10.  The  diameter  of  the  base  of  a  cylinder  is  84  yards, 
and  the  altitude  21  feet:  how  many  solid  or  cubic  yards 
does  it  contain?  Ans.  38792.4768. 


OF    THE    CONE. 

36.  How   is  a   cone  described?      What  is  its  base?  what 
its  convex  surface  ?  what  its  altitude,  and  what  its  vertex  ? 

A   cone  is   a   solid,  described  by  the  ^ 

revolution    of    a     right-angled    triangle 
ABC,  about  one  of  its  sides   CB. 

The  circle  described  by  the  revolving  i 

side  AB,  is  called  the  base  of  the  cone.  M 

The   hypothenuse   AC  is    called   the  It 

slant  height  of  the  cone,  and  the  surface  m'£ 
described  by  it  is  called  the  convex  sur-  A^mm; 
face  of  the  cone.  *~ 


164 


BOOK    V. SECTION    II. 


The  side  of  the  triangle  CB,  which  remains  fixed,  is 
called  the  axis  or  altitude  of  the  cone,  and  the  point  O 
the  vertex  of  the  cone. 


37-   What  is  the  frustum  of  a  cone  ? 

If  a  cone  be  cut  by  a  plane  paral- 
lel to  the  base,  the  section  will  be  a 
circle.  Thus,  the  section  FKHI  is  a 
circle.  If  from  the  cone  S — CDB, 
the  cone  S — FKH  be  taken  away,  the 
remaining  part  is  called  the  frustum  ^ 
of  a  cone. 


38.  How  do  you  find  the  surface  of  a  cone  1 
The    convex   surface   of  a    cone    is 

equal  to  the  circumference  of  the  base 

multiplied   by   half    the    slant    height. 

Thus,  the  convex  surface  of  the  cone 

C — AED  is  equal  to 

circumference  AED  x  I CA  : 

to  this  add  the  area  of  the  base,  when 

the  entire  surface  is  required. 


EXAMPLES. 


1.  What  is  the  convex  surface  of 
the  cone  whose  vertex  is  C,  the  diam- 
eter AD  of  its  base  being  8^  feet,  and 
the  side  CA  50  feet? 


OF    THE    CONE. 


165 


First,     3.1416  x8i  =  26.7036  =  circum.  of  base. 
Then,    — ■ =  667.59  =  convex  surface. 


2.  Required  the  entire  surface  of  a  cone,  whose  side  is 
36,  and  the  diameter  of  its  base  18  feet. 

Ans.  1272.348  sq.ft. 

3.  The  diameter  of  the   base  is   3  feet,   and  the    slant 
height  15  feet:  what  is  the  convex  surface  of  the  cone? 

Ans.  70.686  sq.ft. 

4.  The  diameter  of  the  base  of  a  cone  is  4.5  feet,  and 
the  slant  height  20  feet:   what  is  the  entire  surface? 

Ans.   157.27635  sq.ft. 

5.  The  circumference   of  the  base  of  a   cone  is  10.75 
and  the  slant  height  is  18.25:  what  is  the  entire  surface? 

Ans.  107.29021  sq.ft. 


39.  How  do  you  find  the  solidity  of 
a  cone  ? 

The  solidity  of  a  cone  is  equal  to 
the  area  of  the  base  multiplied  by  one- 
third  of  the  altitude.  Thus,  the  so- 
lidity of  the  cone  C — AED  is  equal  to 
base  AED  x  ±CB. 


40.  How  do  a  cone  and  cylinder,  of  the  same  base  and 
altitude,  compare  with  each  other? 

Since  the  solidity  of  a  cylinder  is  equal  to  the  base 
multiplied  by  the  altitude,  and  that  of  a  cone  to  the  base 
multiplied  by  one-third  of  the  altitude,  it  follows  that  if  a 
cylinder  and  cone  have  equal  bases  and  altitudes,  the  cone 
will  be  one-third  of  the  cylinder. 


166 


BOOK    V. — SECTION    IT 


EXAMPLES. 


I.  What  is  the  solidity  of  a  cone,  the 
area  of  whose  base  is  380  square  feet, 
and  altitude  CB  48  feet? 


We  simply  multiply  the 
area  of  the  base  by  the  al- 
titude, and  then  divide  the 
product  by  3. 


Operation. 

380 
48 


3040 
1520 
3)18240 
area  =  6080 


2.  Required  the  solidity  of  a  cone  whose  altitude  is  27 
feet,  and  the  diameter  of  the  base  10  feet.* 

Ans.  706.86  cubic  ft. 


3.  Required  the  solidity  of  a  cone  whose  altitude  is  10£ 
feet,  and  the  circumference  of  its  base  9  feet. 

Ans.  22.5609  cubic  ft. 

4.  What  is  the  solidity  of  a  cone,  the  diameter  of  whose 
base  is  18  inches,  and  altitude  15  feet? 

Ans.  8.83575  cubic  ft. 

5.  The  circumference  of  the  base  of  a  cone  is  40  feet, 
and  the  altitude  50  feet:   what  is  the  solidity? 

Ans.  2122.1333  solid  ft. 


OF    THE    FRUSTUM    OF    A    CONE. 


167 


OF    THE    FRUSTUM    OF    A    CONE. 

41.  How  do  you  find  the  surface  of  the  frustum  of  a  cone? 

Add  together  the  circumferences  of  the  two  bases,  and 
multiply  the  sum  by  half  the  slant  height  of  the  frustum ; 
the  product  will  be  the  convex  surface,  to  which  add  the 
areas  of  the  bases,  when  the  entire  surface  is  required 


EXAMPLES. 

1.  What  is  the  convex  surface  of  the 
frustum  of  a  cone,  of  which  the  slant 
height  is  12£  feet,  and  the  circumferences 
of  the  bases  8.4  and  6  feet? 


We  merely  take  the  sum 
of  the  circumferences  of  the 
bases,  and  multiply  by  half 
the  slant  height. 


Operation. 

8.4 
6 

half  side     6.25 

area  =  90  sq.ft. 


2.  What  is  the  entire  surface  of  the  frustum  of  a  cone, 
the  side  being  16  feet,  and  the  radii  of  the  bases  2  and 
3  feet? 

Ans.  292.1688  sq.ft. 

3.  What  is  the  convex  surface  of  the  frustum  of  a  cone, 
the  circumference  of  the  greater  base  being  30  feet,  and 
of  the  less   10  feet;  the  slant  height  being  20  feet? 

Ans.  400  sq.  ft. 

4.  Required  the  entire  surface  of  the  frustum  of  a  cone 
whose  slant  height  is  20  feet,  and  the  diameters  of  the 
bases  8  and  4  feet. 

Ans.  439.824  sq,  ft. 


168 


BOOK    V. SECTION    II. 


5.  A  cone  whose  slant  height  is  30  feet,  and  the  circum- 
ference of  its  base  10  feet,  is  cut  by  a  plane  6  feet  from 
the  vertex,  measured  on  the  slant  height :  what  is  the  con- 
vex surface  of  the  frustum  ? 

Ans.  144  sq.  ft. 

42.  How  do  you  find  the  solidity  of  the  frustum  of  a  cone  ? 

1st.  Add  together  the  areas  of  the  two  ends  and  a  geo- 
metrical mean  between  them. 

2d.  Multiply  this  sum  by  one-third  of  the  altitude,  and 
the  product  will  be  the  solidity 


EXAMPLES. 

1.  How  many  cubic  feet  in  the  frus- 
tum of  a  cone,  whose  altitude  is  26  feet, 
and  the  diameters  of  the  bases  22  and 
18  feet? 

First,  222  x  .7854  =  380.134  =  area 

of  lower  base : 

— s 

and  18   x  .7854  =  254.47  =  area  of  upper  base. 

Then,  V380.134  x  254.47  =  311.018  =  mean. 

26 
Then,  (380.134  -J-  254.47  +  311.018)  X  -^-  =  8195.39 

0 

which  is  the  solidity. 

2.  How  many  cubic  feet  in  a  piece  of  round  timber,  the 
diameter  of  the  greater  end  being  18  inches,  and  that  of 
the  less  9  inches,  and  the  length  14.25  feet? 

Ans.  14.68943. 

3.  What  is  the  solidity  of  the  frustum  of  a  cone,  the 
altitude  being  18,  the  diameter  of  the  lower  base  8,  and 
that  of  the  upper  base  4?  Am    527>7888, 


OF    THE    SPHERE. 


169 


4.  What  is  the  solidity  of  the  frustum  of  a  cone,  the 
altitude  being  25,  the  circumference  of  the  lower  base  20, 
and  that  of  the  upper  base  10?  .        464  216 

5.  If  a  cask,  which  is  composed  of  two  equal  conic 
frustums  joined  together  at  their  larger  bases,  have  its  bung 
diameter  28  inches,  the  head  diameter  20  inches,  and  the 
length  40  inches,  how  many  gallons  of  wine  will  it  con- 
tain, there  being  231  cubic  inches  in  a  gallon? 

Ans.  79.0613. 


OF    THE    SPHERE. 

43.   What  is  a  sphere  ? 

A  sphere  is  a  solid  terminated  by  a  curved  surface,  all 
the  points  of  which  are  equally  distant  from  a  certain  point 
within  called  the  centre. 


44.  How  may  a  sphere  be  de- 
scribed ? 

The  sphere  may  be  described  by 
revolving  a  semicircle  ABD  about 
the  diameter  AD.  The  plane  will 
describe  the  solid  sphere,  and  the 
semi-circumference  ABD  will  de- 
scribe the  surface. 


45.  What  is  the  radius  of  a  sphere  ? 

The  radius  of  a  sphere  is  a  line 
drawn  from  the  centre  to  any  point 
of  the  circumference.  Thus,  CA  is 
a  radius  *. 


8 


170 


BOOK    V. SECTION    IT. 


46.  What  is  the  diameter  of  a 
sphere  ? 

The  diameter  of  a  sphere  is  a 
line  passing  through  the  centre,  and 
terminated  by  the  circumference. 
Thus,  AD  is  a  diameter. 


47.  How  do  the  diameters  of  a  sphere  compare  with  each 
other  ? 

All  diameters  of  a  sphere  are  equal  to  each  other ;  and 
each  is  double  a  radius. 

48.  What  is  the  axis  of  a  sphere  ? 

The  axis  of  a  sphere  is  any  line  about  which  it  re- 
volves; and  the  points  at  which  the  axis  meets  the  sur- 
face, are  called  the  poles. 

49.  When  is  a  plane  said  to  be 
tangent  to  a  sphere  ? 

A  plane  is  tangent  to  a  sphere 
when  it  has  but  one  point  in  com- 
mon with  it.  Thus,  AB  is  a  tan- 
gent plane. 

50.  What  is  a  spherical  zone?    what  are  its  bases? 
A  zone  is  a  portion  of  the  surface 

ot  a  sphere,  included  between  two 
parallel  planes  which  form  its  bases. 
Thus,  the  part  of  the  surface  inclu- 
ded between  the  planes  AE  and  DF 
is  a  zone.  The  bases  of  this  zone 
are  two  circles  whose  diameters  are 
AE  and  DF. 


OF    THE    SPHERE. 


171 


51.   When  will  a  zone  have  but  one  base? 

One  of  the  planes  which  bound 
a  zone  may  become  tangent  to  the 
sphere,  in  which  case  the  zone  will 
have  but  one  base.  Thus,  if  one 
plane  be  tangent  to  the  sphere  at  A, 
and  another  plane  cut  it  in  the  cir- 
cle DF,  the  zone  included  between 
them  will  have  but  one  base. 


52.  What  is  a  spherical  segment? 

A  spherical  segment  is  a  portion  of  the  solid  sphere  in- 
cluded between  two  parallel  planes.  These  parallel  planes 
are  its  bases.  If  one  of  the  planes  is  tangent  to  the  sphere, 
the  segment  will  have  but  one  base. 

53.  What  is  the  altitude  of  a  zone  ? 

The  altitude  of  a  zone  or  segment,  is  the  distance  be- 
tween the  parallel  planes  which  form  its  bases. 

54.  How  does  a  plane  cut  a  sphere  ? 

Every  plane  passing  through  a  sphere  intersects  the  solid 
sphere  in  a  circle,  and  the  surface  of  the  sphere  in  the  cir- 
cumference of  a  circle. 


55.  When  does  a  plane  cut  a  sphere  in  a  great  circle? 
when  in  a  small  circle  ? 

If  the  intersecting  plane  passes  through  the  centre  of 
the  sphere,  the  circle  is  called  a  great  circle.  If  it  does 
not  pass  through  the  centre  the  circle  of  section  is  called 
a  small  circle. 


172 


BOOK    V. SECTION    II. 


56.  How  do  you  find  the  surface 
of  a  sphere  ? 

The  surface  of  a  sphere  is  equal 
to  the  product  of  its  diameter  by 
the  circumference  of  a  great  circle 
Thus,  the  surface  of  the  sphere 
whose  centre  is  C,  is  equal  to 
circumference  ABBE  x  AD. 


EXAMPLES 


1.  What  is  the  surface  of  the  sphere 
whose  centre  is  C,  the  diameter  being 
7  feet? 

Ans.  153.9384  sq.ft. 


2.  What   is   the   surface   of   a    sphere   whose   diameter 
is  24?  Ans.   1809.5616. 

3.  Required  the  surface  of  a  sphere  whose  diameter  is 
7921  miles.  Ans.  197111024  sq.  miles. 

4.  What  is  the   surface  of  a   sphere  the  circumference 
of  whose  great  circle  is  78.54?  Ans.   1963.5. 

5.  What  is  the  surface  of  a   sphere  whose   diameter  is 
11  feet? 


Ans.  5.58506  sq.ft. 


57.   How  do  you  find    the  solidity 
of  a  sphere  ? 

The  solidity  of  a  sphere  is  equal 
to  its  surface  multiplied  by  one-third 
of    the    radius.     Thus,    the     sphere 
whose  centre  is   C,  is  equal  to 
surface  x  £  CA. 


OF  THE  SPHERE. 


173 


EXAMPLES. 

1 .  What  is  the  solidity  of  a  sphere 
whose  diameter  is  12  feet? 

First,     3.1416  X  12  =  37.6992  == 
circumference  of  sphere, 

diameter        =  12 


surface  =  452.3904 

one-third  radius  =  2 


solidity 


=  904.7808  cubic  feet. 


2.  The  diameter  of  a  sphere  is  7957.8 :  what  is  its  so* 
lldlty?  Ans.  263863122758.4778. 

3.  The  diameter  of  a  sphere  is  24  yards :  what  is  its 
solid  content? 

Ans.  7238.2464  cubic  yds. 

4.  The  diameter  of  a  sphere  is  8 :  what  is  its  solidity  ? 

Ans.  268.0832. 

58.  What  is  a  second  method  of  finding  the  solidity  of  a 
sphere  ? 

Cube  the  diameter  and  multiply  the  number  thus  found 
by  the  decimal  .5236,  and  the  product  will  be  the  solidity. 


EXAMPLES. 

1.  What  is  the  solidity  of  a  sphere  whose  diameter  is  20  ? 

Ans.  4188.8. 

2.  What  is  the  solidity  of  a  sphere  whose  diameter  is  6  ? 

Ans.  113.0976. 

3.  What  is  the  solidity  of  a  sphere  whose  diameter  is  10* 

Ans.  523.6. 


174 


BOOK    V. SECTION    II. 


OF    SPHERICAL    ZONES. 


59.  How   do  you  find  the   convex  surface   of  a  spherical 


zone  s 


Multiply  the  height  of  the  zone  by  the  circumference  of 
a  great  circle  of  the  sphere,  and  the  product  will  be  the 
convex  surface. 


EXAMPLES 


1.  What  is  the  convex  surface  of 
the  zone  ABD,  the  height  BE  being 
9  inches,  and  the  diameter  of  the 
sphere  42  inches  ? 


First,     42  X  3.1416  =     131.9472  =  circumference, 
height  s±  9 

surface  =  1187.5248  square  inches. 

2.  The  diameter  of  a  sphere  is  12£  feet:  what  will  be 
the  surface  of  a  zone  whose  altitude  is  2  feet? 

Ans.  78.54  sq.ft. 

3.  The  diameter  of  a  sphere  is  21  inches :   what  is  the 
surface  of  a  zone  whose  height  is  4£  inches  ? 

Ans.  296.8812  sq.  in. 

4.  The  diameter  of  a  sphere  is  25  feet,  and  the  height 
of  the  zone  4  feet :  what  is  the  surface  of  the  zone  ? 

Ans.  314.16  sq.ft. 


OF    SPHERICAL    SEGMENTS. 


60.  How  do  you  find  the  solidity  of  a  spherical  segment 
with  one  base? 

1st.  To  three  times  the  square  of  the  radius  of  the  base 
add  the  square  of  the  height. 


OF    SPHERICAL    SEGMENTS.  175 

2d.  Multiply  this  sum  by  the  height,  and  the  product  by 
the  decimal  .5236 ;  the  result  will  be  the  solidity  of  the 
segment. 

EXAMPLES. 

1.  What  is  the  solidity  of  the  seg- 
ment ABD,  the  height  BE  being 
4  feet,  and  the  diameter  AD  of  the 
base  being  14  feet? 

First, 

(72  x  3  +  I2) •  =  147  +  16  =  163. 

Then,  163  x  4  x  .5236  ==  341.3872  solid  feet,  which  is 
the  solidity  of  the  segment. 

2.  What  is  the  solidity  of  the  segment  of  a  sphere, 
whose  height  is  4,  and  the  radius  of  its  base  8? 

Ans.  435.6352. 

3.  What  is  the  solidity  of  a  spherical  segment,  the  di- 
ameter of  its  base  being  17.23368,  and  its  height  4.5  ? 

Arts.  572.5566. 

4.  What  is  the  solidity  of  a  spherical  segment,  the  di- 
ameter of  the  sphere  being  8,  and  the  height  of  the  seg- 
ment 2  feet? 

Ans.  41.888  cubic  ft. 

5.  What  is  the  solidity  of  a  segment,  when  the  diameter 
of  the  sphere  is  20,  and  the  altitude  of  the  segment  9  feet  ? 

Ans.  178J.2872  cubic  ft. 

61.  How  do  you  find  the  solidity  of  a  spherical  segment 
having  two  bases  ? 

To  the  sum  of  the  squares  of  the  radii  of  the  two  bases 
add  one-third  of  the  square  of  the  distance  between  them ; 
then  multiply  this  sum  by  the  breadth,  and  the  product  by 
1.5708,  and  the  result  will  be  the  solidity. 


176 


BOOK    V.  — SECTION    II. 


EXAMPLES. 

1.  What  is  the  solid  content  of  the 
zone  ADFE,  the  diameter  of  whose 
greater  base  DF  is  equal  to  20  inches, 
and  the  less  diameter  AE  15  inches, 
and  the  distance  between  the  two  ba- 
ses 10  inches? 

Now,  by  the  rule 

[(10)2  +  (7.5)8+^]  X  10  X  1.5708 

=  (100  +  56.25  +  33.33)  X  10  X  1.5708 

=z  189.58  X  10  x  1.5708  =  2977.92264  solid  inches. 

2.  What  is  the  solid  content  of  a  zone,  the  diameter  of 
whose  greater  base  is  24  inches,  the  less  diameter  20 
inches,  and  the  distance  between  the  bases  4  inches  ? 

Ans.  1566.6112  solid  in. 

3.  What  is  the  solidity  of  the  middle  zone  of  a  sphere, 
the  diameter  of  whose  bases  are  each  3  feet,  and  the  dis- 
tance between  them  4  feet?  Ans.  61.7848  solid  ft. 


OF    THE    SPHEROID. 

62.  What  is  a  spheroid? 

A  spheroid  is  a  solid,  described  by  the  revolution  of  an 
ellipse  about  either  of  its  axes. 

63.  What  is  the  difference  between  a  prolate  and  an  oblate 
spheroid  ? 

If  an  ellipse  ACBD  be  re-  ^ 

volved  about  the  transverse  or 
longer  axis  AB,  the  solid  de- 
scribed is  called  a  prolate 
spheroid ;  and  if  it  be  revolved 
about   the   shorter  axis    CD,  c 

the  solid  described  is  called  an  oblate  spheroid. 


OF    THE    SPHEROID.  177 

64.  What  is  the  form  of  the  earth  1 

The  earth  is  an  oblate  spheroid,  the  axis  about  which 
it  revolves  being  about  34  miles  shorter  than  the  diameter 
perpendicular  to  it. 

65.  How  do  you  find  the  solidity  of  an  ellipsoid? 
Multiply  the  fixed  axis  by  the  square  of  the  revolving 

axis,  and  the   product   by  the   decimal   .5236 ;    the   result 
will  be  the  required  solidity. 

EXAMPLES. 

1.  In  the  prolate  spheroid 
ACBD,  the  transverse  axis 
AB  =  90,  and  the  revolving 
axis  CD  =  70  feet :  what  is 
the  solidity  ? 

Here,    AB  =  90  feet :    CD*  =  70a  =  4900  :  hence 

AB  X  CD*  X  .5236  =  90  x  4900  x  .5236  =s  230907.6 
cubic  feet,  which  is  the  solidity. 

2.  What  is  the  solidity  of  a  prolate  spheroid,  whose 
fixed  axis  is  100,  and  revolving  axis  6  feet? 

Ans.  1884.96. 

3.  What  is  the  solidity  of  an  oblate  spheroid,  whose  fixed 
axis  is  60,  and  revolving  axis  100? 

Ans.  314160. 

4.  What  is  the  solidity  of  a  prolate  spheroid,  whose  axes 
are  40  and  50?  Ans.  41888. 

5.  What  is  the  solidity  of  an  oblate  spheroid,  whose  axes 
are  20  and  10?  Ans.  2094.4. 

6.  What  is  the  solidity  of  a  prolate  spheroid,  whose  axes 
are  55  and  33?  Ans.  31361.022. 

8* 


176 


BOOK    V. SECTION    II. 


OF     CYLINDRICAL    RINGS. 

66.  How  is  a  cylindrical  ring  formed  1 

A  cylindrical  ring  is  formed  by- 
bending  a  cylinder  until  the  two 
ends  meet  each  other.  Thus,  if  a 
cylinder  be  bent  round  until  the 
axis  takes  the  position  mon,  a  solid 
will  be  formed,  which  is  called  a 
cylindrical  ring. 

The  line  AB  is  called  the  outer,  and  cd  the  inner  di- 
ameter. 

67.  How  do  you  Jind  the  convex  surface  of  a   cylindrical 
ring  ? 

1st.  To  the  thickness  of  the  ring  add  the  inner  diameter. 
2d.  Multiply  this  sum  by  the  thickness,  and  the  product 
by  9.8696 ;  the  result  will  be  the  area. 


EXAMPLES. 

1.  The  thickness  Ac  of  a  cylin- 
drical ring  is  3  inches,  and  the  in- 
ner diameter  cd  is  12  inches :  what 
is  the  convex  surface  ? 

Ac  +  cd  =  3  +  12  =  15 :  then 
15  x  3  x  9.8696  =  444.132  square 
inches  =  the  surface. 

2.  The  thickness  of  a  cylindrical  ring  is  4  inches,  and 
the  inner  diameter  18  inches:  what  is  the  convex  surface? 

Ans.  868.52  sq.  in. 

3.  The  thickness  of  a  cylindrical  ring  is  2  inches,  and 
the  inner  diameter  18  inches:  what  is  the  convex  surface? 

Ans.  394.784  sq.  in. 


OF    THE    FIVE    REGULAR    SOLIDS.  179 

68.  How  do  you  find  the  solidity  of  a  cylindrical  ring  ? 

1st.  To  the  thickness  of  the  ring  add  the  inner  diam- 
eter. 

2d.  Multiply  this  sum  by  the  square  of  half  the  thick- 
ness, and  the  product  by  9.8696 ;  the  result  will  be  the 
required  solidity. 

EXAMPLES. 

1.  What  is  the  solidity  of  an  anchor-ring,  whose  inner 
diameter  is  8  inches,  and  thickness  in  metal  3  inches? 

8  +  3  =  11  :  then,  11  x  (f)2  X  9.8696  =  244.2726,  which 
expresses  the  solidity  in  cubic  inches. 

2.  The  inner  diameter  of  a  cylindrical  ring  is  18  inches, 
and  the   thickness  4  inches :    what  is  the  solidity  of  the 

lmg  "  Ans.  868.5248  cubic  in. 

3.  Required  the  solidity  of  a  cylindrical  ring,  whose 
thickness  is  2  inches,  and  inner  diameter  12  inches? 

Ans.   138.1744  cubic  in. 

4.  What  is  the  solidity  of  a  cylindrical  ring,  whose 
thickness  is  4  inches,  and  inner  diameter  16  inches? 

Ans.  789.568  cubic  in 


OF    THE    FIVE    REGULAR    SOLIDS. 

69.  A  regular  solid  is  one  whose  faces  are  all  equal  poly 
gons,  and  whose  solid  angles  are  equal.  There  are  fivo 
such  solids. 


m 


180 


BOOK    V. — SECTION    II. 


1.    The    tetraedron,   or    equilateral   pyramid,   is    a    solid 
bounded  by  four  equal  triangles. 


Pyramid  unfolded. 


Pyramid. 


2.  The   hexaedron,  or  cube,  is  a   solid  bounded  by   six 
equal  squares. 


t^---::,  P";~"    ! 


Cube  unfolded 


Cube. 


3.  The  octaedron,  is  a  solid  bounded  by  eight  equal  tri- 
angles. 


Octaedron  unfolded. 


Octaedron 


OF    THE    FIVE    REGULAR    SOLIDS. 


181 


4.  The  dodecaedron  is  a  solid  bounded  by  twelve  equal 
pentagons. 


Dodecaedron  unfolded. 


Dodecaedron 


5.  The   icosaedron   is  a  solid  bounded  by  twenty  equal 
triangles. 


Icosaedron  unfolded. 


Icosaedron. 


6.  The  regular  solids  may  easily  be  made  of  paste- 
board , 

Draw  the  figures  of  the  unfolded  regular  solids  accurately  on 
pasteboard,  and  then  cut  through  the  bounding  lines :  this 
will  give  figures  of  pasteboard  similar  to  the  diagrams. 
Then,  cut  the  other  lines  half  through  the  pasteboard ;  after 
which,  fold  up  the  parts,  and  glue  them  together,  and  you 
will  form  the  bodies  which  have  been  described. 


182 


BOOK    V. SECTION    II. 


The   following  table   shows  the   surface  and  solidity  of 
each  of  the  regular  solids,  when  the  linear  edge  is  unity. 


No.  of  sides.                   Names. 

Surfaces. 

Solidities. 

4 

Tetraedon 

1.73205 

0.11785 

6 

Hexaedron 

6.00000 

1.00000 

8 

Octaedron 

3.46410 

0.47140 

12 

Dodecaedron 

20.64578 

7.66312 

20 

Icosaedron 

8.66025 

2.18169 

69.  How  will  you  find  the  surface  of  a  regular  solid,  when 
the  length  of  the  linear  edge  is  given  ? 

Multiply  the  square  of  the  linear  edge  by  the  tabular 
number  in  the  column  of  surfaces,  and  the  product  will  be 
the  surface  required. 

EXAMPLES. 

1.  The  linear  edge  of  a  tetraedron  is  3:  what  is  its 
surface  1 

The  tabular  area  is  1.73205.     Then, 

32  =  9 ;  and  1.73205  x  9  =  15.58845  =  surface. 

2.  The  linear  edge  of  an  octaedron  is  5 :  what  is  its 
surface  1  W 

The  tabular  area  is  3.46410.     Then, 

5a  =  25  ;  and  3.46410  X  25  =  86.6025    =  surface. 

3.  The  linear  edge  of  an  icosaedron  is  6:  what  is  its 
surface  1 

The  tabular  area  is  8.66025.     Then, 

6s  =  36 ;  and  8.66025  x  36  =  311.769  =  surface. 


OF    THE    FIVE    REGULAR    SOLIDS.  183 

70.  How  do  you  find  the  solidity  of  a  regular  solia\  when 
the  length  of  a  linear  edge  is  known  ? 

Multiply  the  cube  of  the  linear  edge  by  the  tabular  num- 
ber in  the  column  of  solidities,  and  the  product  will  be  the 
solidity  required. 

EXAMPLES. 

1.  What  is  the  solidity  of  a  regular  tetraedron  whose 
side  is  6  ? 

The  tabular  number  in  the  column  of  solidities  is  0.11785. 
Then, 

63  =  216  ;  and  0.11785  X  216  =  25.4556. 

2.  What  is  the  solidity  of  a  regular  octaedron  whose 
linear  edge  is  8? 

The  tabular  number  in  the  column  of  solidities  is  0.47140, 
Then, 

8s  =  512;  and  0.47140  X  512  =  241.35680  =  solidity. 

3.  What  is  the  solidity  of  a  regular  dodecaedron  whose 
linear  edge  is  3  1 

The  tabular  number  in  the  column  of  solidities  is  7.66312. 
Then, 

33  =  27  ;  and  7.66312  X  27  =  206.90424  =  solidity. 

4.  What  is  the  solidity  of  a  regular  icoM,edron  whose 
linear  edge  is  3  1 

The  tabular  number  in  the  column  of  solidities  is  2  18169 
Then, 

3*  =  27;  and  2.18169  X  27  =  58.90563  =  solidity. 


184  BOOK    VI. — SECTION    I. 


BOOK  VI. 


ARTIFICERS'    WORK. 
SECTION   I. 

OF    MEASURES. 

1.  What  is  the  carpenter's  rule  used  for? 

The  carpenter's  rule,  sometimes  called  the  sliding  rule, 
is  used  for  the  measurement  of  timber,  and  artificers'  work. 
By  it  the  dimensions  are  taken,  and  by  means  of  certain 
scales,  the  superficial  and  solid  contents  may  be  computed. 

2.  Describe  the  rule . 

The  rule  consists  of  two  equal  pieces  of  box  wood,  each 
one  foot  long,  and  connected  together  by  a  folding  joint. 

One  face  of  the  rule  is  divided  into  inches,  half  inches, 
quarter  inches,  eighths  of  inches,  and  sixteenths  of  inches 
When  the  rule  is  opened,  the  inches  are  numbered  from  1 
to  23,  the  last  number    24,  at  the  end,  being  omitted. 

3.  How  is  the  edge  of  the  rule  divided  ? 

The  edge  of  the  rule  is  divided  decimally;  that  is,  each 
foot  is  divided  into  ten  equal  parts,  and  each  of  those  again 
into  ten  parts,  so  that  the  divisions  on  the  edge  of  the  scale 
are  hundredths  of  a  foot.  The  hundredths  are  numbered 
on  each  arm  of  the  scale,  from  the  right  to  tho  left. 


OF    MEASURES. 


185 


4.  How  are  inches  changed  to  the  decimal  of  a  foot  ? 

By  means  of  the  decimal  divisions  it  is  easy  to  convert 
inches  into  the  decimal  of  a  foot. 

Thus,  if  we  have  6  inches,  we  find  its  corresponding 
decimal  on  the  edge  of  the  rule  to  be  50  hundredths  of  a 
foot,  or  .50.  Also  9  inches  correspond  to  .75 ;  8  inches  to 
.67  nearly,  and  3  inches  to  .25. 

5.  How  are  feet  and  inches  multiplied  by  means  of  deci- 
mals ? 

The  multiplication  of  numbers  is  more  easily  made  when 
the  numbers  are  expressed  decimally  than  when  expressed 
in  feet  and  inches. 

Let  us  take  an  example.  A  board  is  12  feet  6  inches 
long,  and  2  feet  3  inches  wide :  how  many  square  feet 
does  it  contain? 

We  see  from  the  edge  of  the  rule,  that  6  inches  cor- 
respond to  .50,  and  -3  inches  to  .25.     Hence,  we  have 


By  cross  multiplication. 

By  decimals. 

12  6' 

12.50 

2  3' 

2.25 

25 

6250 

3  V  6" 

2500 

28  V  6"  content. 

2500 

28.1250  content. 

6.    What  are  the  objects  of  the  scale  marked  M  and  E  ? 

Besides  the  scale  of  feet  and  inches,  already  referred 
to,  there  are,  on  the  same  side,  two  small  scales,  marked 
M  and  E  ;  the  first  is  numbered  from  1  to  36,  and  the  sec- 
ond from  1  to  26.  The  object  of  these  scales  is  to  change 
a  square  into  what  is  called  in  carpentry  an  eight  square, 
or  regular  octagon. 


186  BOOK    VI. — SECTION    I. 

7.  Explain  the  use  of  the  one  marked  M. 

Having  formed  the  square  which  is  to  be  changed  to 
the  octagon,  find  the  middle  of  each  side,  and  then  the 
divisions  of  the  scale  marked  M  show  the  distances  to  be 
laid  off  on  each  side  of  the  centre  points,  to  give  the  an- 
gles of  the  octagon. 

For  example,  if  the  side  of  the  square  is  6  inches,  the 
distance  to  be  laid  off  is  found  by  extending  the  dividers 
from  1  to  6.  If  the  side  of  the  square  is  12  inches,  the 
distance  to  be  taken  reaches  from  1  to  12 ;  and  so  on  for 
any  distance  from  1  to  36. 

8.  Explain  the  use  of  the  one  marked  E. 

The  scale  marked  E  is  for  the  same  object,  only  the 
distances  are  laid  off  from  the  angular  points  of  the  square 
instead  of  from  the  centre. 

Thus,  if  we  have  a  square  whose  side  is  9  inches,  and 
wish  to  change  it  into  an  octagon,  take  from  the  scale  E 
the  distance  from  1  to  9,  and  mark  it  off  from  each  angle 
of  the  square,  on  the  sides :  then  join  the  points,  and  the 
figure  so  formed  will  be  a  regular  octagon. 

If  the  side  of  the  square  is  18  inches,  the  distance  to 
be  taken  reaches  from  1  to  18,  and  so  for  any  distance 
between  1  and  26,  the  numbers  on  the  scale  pointing  out 
the  distances  to  be  laid  off  when  the  side  of  the  square  is 
expressed  in  inches. 

•  9.   What  scales  are  on   the  opposite  face  of  the  rule,  and 
how  are  they  designated? 

Turning  the  rule  directly  over,  there  will  be  seen  on 
one  arm  several  scales  of  equal  parts,  which  are  similar 
to  those  described  at  page  36. 

Fitting  into  the  other  arm  is  a  small  brass  slide,  of  the 
same  length  as  the  rule.     On  the  face  of  the  slide  are  two 


OF    MEASURES.  187 

ranges  of  divisions,  which  are  precisely  alike.  The  upper 
is  designated  by  the  letter  B,  and  is  to  be  used  with  the 
scale  on  the  rule  directly  above,  which  is  designated  by  A ; 
the  lower  divisions  on  the  slide  designated  by  the  letter  C, 
are  to  be  used  with  the  scale  mark  girt  line,  and  also 
designated  by  the  letter  D.  The  scales  B  and  C  on  the 
slide,  are  numbered  1,  2,  3,  4,  5,  6,  7,  8,  9,  and  1,  from 
the  left  hand  towards  the  right.  From  the  middle  point 
1,  the  numbers  go  on  12,  2,  3,  4,  5,  6,  7,  8,  9,  and  10. 
Now,  the  values  which  the  parts  of  this  scale  may  repre- 
sent will  depend  on  the  value  given  to  the  unit  at  the  left 
hand.  If  the  unit  at  the  left  be  called  1,  then  the  1  at  the 
centre  point  will  represent  10,  and  the  2  at  the  right  20, 
the  3  at  the  rjght  30,  and  the  ten  100,  and  similarly  for 
the  intermediate  divisions. 

If  the  left-hand  unit  be  called  10,  then  the  1  at  the 
centre  point  will  represent  one  hundred ;  the  2,  two  hun- 
dred ;  the  3,  three  hundred ;  and  so  on  for  the  divisions 
to  the  right. 

10.  How  do  you  multiply  two  numbers  together  by  the  sli- 
ding rule  1 

1st.  Mark  a  number  on  the  scale  A  to  represent  the 
multiplier. 

2d.  Then  shove  the  slide  until  1  on  5  stands  opposite 
the  multiplier  on  A. 

3d.  Then  pass  along  on  B  until  you  find  a  number  to 
represent  the  multiplicand ;  the  number  opposite  on  A  will 
represent  the  product. 

EXAMPLES. 

1.  Multiply  24  by  14. 

Move  the  slide  unil  1  on  B  is  opposite  the  second  long 
mark  at  the  right  of  12,  which  is  the  division  correspond- 


188  BOOK    VI. SECTION    I; 

ing  to  14.  Then  pass  along  B  to  the  fourth  of  the  larger 
lines  on  the  right  of  2  :  this  line  marks  the  division  on 
the  scale  A,  which  shows  the  product.  Now  we  must  re- 
mark that  the  unit  on  the  product  line  is  always  ten  times 
greater  than  the  unit  1  at  the  left  of  the  slide :  and  since 
in  the  example  this  unit  was  10,  it  follows  that  the  3  on 
A  will  stand  for  300,  and  each  of  the  smaller  divisions  for 
10 ;  hence  the  product  as  shown  by  the  scale  is  nearly 
340,  and  by  judging  by  the  eye,  we  write  it  336. 

2.  What  is  the  product  of  36  by  22  ? 

Move  the  slide  till  1  on  B  stands  at  22  on  A;  then 
pass  along  on  B  to  the  6th  line  between  3  and  4 :  the 
figures  on  A  will  then  stand  for  hundreds,  and  the  pro- 
duct will  be  pointed  out  a  little  to  the  right  of  the  9th 
line,  between  7  and  8 ;    or  it  will  be  792. 

3.  A  board  is  16  feet  9  inches  long,  and  15  inches,  or 
1  foot  and  3  inches  wide  :  how  many  square  feet  does  it 
contain  1 

First,  16  feet  and  9  inches  =  16.75  feet; 
and  15  inches  =  1.25  feet: 
Place  1  on  B  at  the  line  corresponding  to  16,  between 
12  and  2  on  i,  and  then  move  over  three-fourths  of  the 
distance  to  the  next  long  line  to  the  right.  Then  looking 
along  on  A,  one  quarter  of  the  distance  between  1  and  2, 
we  find  the  area  of  the  board  to  be  21  feet,  which  is  cor- 
rect, very  nearly. 

4.  The  length  of  a  board  is  15  feet  8  inches,  and  the 
breadth  1  foot  6  inches:    what  is  the  superficial  content? 

15  feet  8  inches  =  15.7  nearly. 

1  foot  6  inches  ±s     1.5  feet. 

Then,  place  1  on  B  at  15.7  on  A,  and  1   and  a  half  on 

B  will  mark  23  and  a  half  feet  on  A,  which  is  the  area 

very  nearly. 


OF    MEASURES.  189 

11.  Explain  the  manner  in  which  the  girt-liue  is  num- 
bered. 

Below  the  slide,  and  on  the  same  side  with  the  scales 
already  described,  is  a  row  of  divisions  marked  girt-line, 
and  numbered  from  4  to  40.  This  line  is  also  designated 
on  the  scale  by  the  letter  D.  The  object  of  this  girt-line, 
which  is  to  be  used  in  conjunction  with  the  sliding  scale, 
is  to  find  the  solid  content  of  timber. 

12.  What  is  the  quarter-girt,  and  how  do  you  find  it  1 
The  quarter-girt,  as  it  is  called  in  the  language  of  me- 
chanics, is  one  quarter  the  circumference  of  a  stick  of 
timber  at  its  middle  point.  The  quarter-girt,  in  squared 
timber,  is  found  by  taking  a  mean  between  the  breadth  and 
thickness. 

Thus,  if  the  breadth  at  the  middle  point  is  4  feet  6  inches, 
and  the  thickness  3  feet  4  inches,  we  have 

ft.    in. 
4     6  breadth 
3     4  depth 


2)7  10 


3  11  quarter-girt. 


and  hence  the  quarter-girt  is  3  feet  11  inches. 

13.  "When  a  stick  of  timber  tapers  regularly,  how  do  you 
find  the  quarter-girt? 

If  a  stick  of  timber  tapers  regularly  from  one  end  to  the 
other,  the  breadth  and  depth  at  the  middle  point  may  be 
found  by  taking  the  mean  of  the  breadth  and  depth  at  the 
ends. 

Thus,  if  the  breadths  at  the  ends  are  1  foot  6  inches, 
and  1  foot  3  inches,  the  mean  breadth  will  be  1  foot  4£ 
inchos.     And,  if  the  depths  at  the  ends  are  1  foot  3  inches, 


190  BOOK    VI. — SECTION    I. 

and  1  foot,  the  mean  depth  or  thickness  will  be  1  foot  1£ 
inches ;   and  the  quarter-girt  will  be  1  foot  3  inches. 

14.  How  do  you  find,  by  the  sliding  rule,  the  solid  con- 
tent of  a  stick  of  timber.,  when  the  length  and  quarter-girt 
are  known  ? 

1st.  Reduce  the  length  of  the  timber  to  feet  and  deci- 
mals of  a  foot,  and  the  qUarter-girt  to  inches. 

2d.  Note  on  scale  C  the  number  which  expresses  the 
length,  and  move  the  slide  until  this  number  falls  at  12  on 
the  girt-line. 

3d.  Pass  along  on  the  girt-line  till  you  find  the  number 
which  expresses  the  quarter-girt  in  inches,  and  the  division 
which  it  marks  on  C  will  show  the  content  of  the  timber 
in  cubic  feet. 

EXAMPLES. 

1.  A  piece  of  square  timber  is  3  feet  9  inches  broad, 
2  feet  7  inches  thick,  and  20  feet  long:  how  many  solid 
feet  does  it  contain? 

ft.  in. 
3    9 
2    7 


2)6    4 


3    2  quarter-girt  =  38  inches. 


Now,  move  the  slide  until  20  on  C  falls  at  12  on  the 
girt-line.  If  we  take  1  on  C  at  the  left  for  10,  2  will 
represent  20,  which  is  placed  opposite  12  on  D.  Then 
passing  along  the  girt-line  to  division  38,  we  find  the  con- 
tent on  C  to  be  a  little  over  200,  say  200^. 

2.  The  length  of  a  piece  of  timber  is  18  feet  6  inches, 
the  breadths  at  the  greater  and  less  ends  are  1  foot  6 
inches,   and   1   foot   3   inches  ;    and  the   thickness   at   the 


OF    MEASURES.  191 

greater  and  less  end,  1  foot  3  inches  and  1  foot :  what  is 
the  solid  content  ? 

Here,  the  mean  breadth  is  1  foot  4£  inches,  the  mean 
thickness  1  foot  1£  inches,  and  the  quarter-girt  1  foot  3 
inches,  or  15  inches. 

Therefore,  place  18.5  on  C,  at  12  on  D,  and  pass  along 
the  girt-line  to  15;  the  number  on  C,  which  is  a  little 
more  than  28  and  a  half,  will  express  the  solid  content. 

TABLE  FOR  BOARD  MEASURE. 

15.  Explain  the  table  rule  for  finding  the  content  of  boards. 

Besides  the  carpenter's  rule  with  a  slide,  which  we  have 
just  described,  there  is  another  folding  rule  without  a  slide, 
and  on  the  face  of  which  is  a  table  to  show  the  content 
of  a  board  from  1  to  20  feet  in  length,  and  from  6  to  20 
inches  in  width. 

The  upper  line  of  the  table  shows  the  length  of  the 
board  in  feet,  and  the  column  at  the  left  shows  the  width 
of  the  board  in  inches,  from  6  to  20.  For  convenience, 
however,  the  table  is  often  divided  into  two  parts,  which 
are  placed  by  the  side  of  each  other. 

EXAMPLES. 

1.  If  your  board  is  6  inches  wide,  and  14  feet  long,  cast 
your  eye  along  the  top  line  till  you  come  to  14 ;  directly 
under  you  will  find  7,  which  shows  that  the  board  contains 
7  square  feet. 

2.  If  your  board  is  10  inches  wide,  and  16  feet  long, 
cast  your  eye  along  the  top  line  till  you  come  to  16;  then 
pass  along  down  till  you  come  to  the  line  of  10 :  the  num- 
ber thus  found  is  13-4,  which  shows  that  the  board  con- 
tains 13  and  4  twelfths  square  feet. 


192  BOOK    VI. SECTION    I. 

The  right-hand  side  of  the  table  begins  at  13  inches  on 
the  left-hand  column. 

3.  What  is  the  content  of  a  board  which  is  13  feet  long, 
and  1 9  inches  wide  ? 

Look  along  the  upper  line  to  13;  then  descend  to  the 
line  19,  where  you  will  find  the  number  20-7,  which  shows 
that  the  board  contains  20  and  7  twelfths  square  feet. 

4.  If  your  board  is  17  inches  wide,  and  14  feet  long, 
you  will  look  under  14  till  you  come  on  to  the  line  17, 
where  you  will  find  the  number  19-10;  which  shows  that 
the  board  contains  19  and  10  twelfths  square  feet. 

5.  If  you  have  a  board  24  feet  long,  and  20  inches  wide, 
first  take  the  area  for  20  feet  in  length,  and  then  for  4 
feet.     Thus, 

for  20  feet  by  20  inches,     33  4 
for  4  feet  by  20  inches,         6  8 

their  sum  gives     40  0  square  feet. 

Note. — Add  as  above  for  any  different  lengths  or  widths. 

If  your  stuff  is  1^  inches  thick,  add  half  to  it. 

If  2  inches  thick,  you  must  double  it. 

The  table  on  the  four-fold  Rule  is  not  divided. 

BOARD    MEASURE. 

16.  Explain  the  board  measure,  and  the  manner  of  using  it. 

This  is  a  measure  two  feet  in  length,  of  an  octagonal 
form,  that  is,  having  eight  faces. 

On  the  line  running  round  the  measure,  at  the  centre, 
we  find  the  faces  of  the  measure  marked,  in  succession, 
by  the  figures  8,  9,  10,  11,  12,  13,  14,  and  15;  and  wc 
shall  designate  each  face  by  the  figure  which  thus  marks 
it.     We  will  likewise  observe,  that  figures   corresponding 


OF    TIMBER    MEASURE.  193" 

to  these,  are  also  sometimes  placed  at  one  end  of  the 
measure. 

Now,  these  figures  at  the  centre  of  the  measure  corre- 
spond to  the  length  of  the  board  to  be  measured.  Thus, 
if  the  board  were  13  feet  in  length,  place  the  thumb  on 
the  line  13  at  the  centre,  and  then  apply  the  measure 
across  the  board,  and  the  number  on  the  face  13,  which 
the  width  of  the  board  marks,  will  express  the  number  of 
square  feet  in  the  board.  Thus,  if  the  width  of  the  board 
extended  from  1  to  10,  the  board  would  contain  15  square 
feet. 

If  the  board  to  be  measured  was  14  feet  long,  its  con- 
tent would  be  measured  on  face  14.  If  the  board  were 
18  feet  long,  measure  its  width  on  face  8,  and  also  on 
face  10,  and  take  the  sum  for  the  true  content  of  the 
board. 

The  Measures  described  above,  are  made  by  Jones  &  Co.,  of  Hartford,  Cu 


SECTION   II. 


OF    TIMBER    MEASURE. 


1.  What  methods  have  already  been  explained? 

The  methods  of  finding  both  the  superficial  content  of 
boards  and  the  solid  content  of  timber,  by  rules  and  scales, 
have  already  been  given.  We  shall  now  give  the  more 
accurate  methods  by  means  of  figures. 

2.  How  do  you  Jind  the  area  of  a  board,  or  plank? 
Multiply  the  length  by  the  breadth,  and  the  product  will 

be  the  content  required. 

9 


194  BOOK    Vl". — SECTION    II. 

3.  How  do  you  find  it  when  the  board  tapers  ? 

If  the  board  is  tapering,  add  the  breadths  of  the  two  ends 
together,  and  take  half  the  sum  for  a  mean  breadth,  and 
multiply  the  result  by  the  length. 

4.  How  may  the  examples  be  done  ? 

The  examples  may  either  be  done  by  cross  multipiica 
tion,  or  the  inches  may  be  reduced  to  the  decimals  of  a 
foot,  and  the  numbers  then  multiplied  together. 

EXAMPLES. 

1.  What  is  the  area  of  a  board  whose  length  is  8  feet 
6  inches,  and  breadth  1  foot  3  inches? 


By  cross  multiplication, 
ft.  in. 
8    6 

1  3 
8~6/ 

2  1    6" 


10    7;  6"  content. 


By  decimals, 
ft.  in. 

8    6   =8.  5  ft. 
1    3' =  1.25' 
Product  =  10.625  sq.ft. 


2.  What  is  the  content  of  a  board  12  feet  6  inches  long, 
and  2  feet  3  inches  broad? 

Ans.  28  ft.  V  6",  or  28.125  sq.ft.     ' 

3.  How  many  square  feet  in  a  board  whose  breadth  at 
one.  end  is  15  inches,  at  the  other  17  inches,  the  length 
of  the  board  being  6  feet?  *        g 

4.  How  many  square  feet  in  a  plank  whose  length  is 

20  feet,  and  mean  breadth  3  feet  3  inches? 

Ans.  65. 

5.  What  is  the  value  of  a  plank  whose  breadth  at  one 
end  is  2  feet,  and  at  the  other  4  feet,  the  length  of  the 
plank  being  12  feet,  and  the  value  per  square  foot  10  cents  1 

Ans.  $3.60 


OF    TIMBER    MEASURE.  195 

5  Having  given  one  dimension  of  a  plank  or  boards  how 
do  you  jind  the  other  dimension  such,  that  the  plank  shall 
contain  a  given  area? 

Divide  the  given  area  by  the  given  dimension,  and  the 
quotient  will  be  the  other  dimension. 

EXAMPLES. 

1.  The  length  of  a  board  is  16  feet;  what  must  be  its 
width  that  it  may  contain  12  square  feet? 

16  feet  =192  inches 

12  square  feet  =  144  x  12  =  1728  square  inches. 
Then,   1728  -r-  192  =  9  inches,  the  width  of  the  board. 

2.  If  a  board  is  6  inches  broad,  what  length  must  be 
cut  from  it  to  make  a  square  foot?  .        9   r. 

3.  If  a  board  is  8  inches  wide,  what  length  of  it  will 
make  4  square  feet?  .        fi   r. 

4.  A  board  is  5   feet  3   inches  long;    what  width  will 

make  7  square  feet?  „        ,    n   A  ■ 

u  Ans.   1  jt.  4  in. 

5.  What  is  the  content  of  a.  board  whose  length  is  5 
feet  7  inches,  and  breadth  1  foot  10  inches? 

Ans.  10  ft.  2'  10". 

6.  How  do  you  find  the  solid  content  of  squared  or  four, 
sided  timber  which  does  not  taper? 

Multiply  the  breadth  by  the  depth,  and  then  multiply  the 
product  by  the  length :  the  result  will  be  the  solid  content. 

EXAMPLES. 

1.  A  squared  piece  of  timber  is  15  inches  broad,  15 
inches  deep,  and  18  feet  long:  how  many  solid  feet  does 

h  COntain?  A*,.  2S.125. 


196  HOOK    VI. SECTION    II. 

2.  What  is  the  solid  content  of  a  piece  of  timber  whose 
breadth  is  16  inches,  depth  12  inches,  and  length  12  feet? 

Ans.  16  ft. 

3.  The  length  of  a  piece  of  timber  is  24.5  feet;  its  ends 
are  equal  squares^whose  sides  are  each  1 .04  feet :  what 
is  the  solidity? 

Ans.  26.4992  solid  ft. 

7.  How  do  you  find  the  solidity  of  a  squared  piece  of  tim 
her  which  tapers  regularly  1 

1st.  Add  together  the  breadths  at  the  two  ends,  and  also 
the  depths., 

2d.  Multiply  these  sums  together,  and  to  the  result  add 
the  products  of  the  depth  and  breadth  at  each  end. 

3d.  Multiply  the  last  result  by  the  length,  and  take  one 
sixth  of  the  product,  which  will  be  the  solidity. 

EXAMPLES. 

1.  How  many  cubic  feet  in  a  piece  of  timber  whose 
ends  are  rectangles,  the  length  and  breadth  of  the  larger 
being  14  inches  and  12  inches;  and  of  the  smaller,  6  and 
4  inches,  the  length  of  the  piece  being  30£  feet? 

14  12  16  X  20  =  320 

_6  _4  14  x  12  =  168 

20  16  6x4  =    24 

512  square  inches. 

But,  512  square  inches  =  *£   square  feet. 
Then,     V  X  301  x  }  =  18^  solid  feet. 

2.  How  many  solid  inches  in  a  mahogany  log,  the  depth 

and  breadth  at  one  end  being  81 1  inches  and  55  inches, 

and  of  the  other  41  and  29£  inches,  the  length  of  the  log 

being  47£  inches? 

Ans.  126340.59375. 


OF    TIMBER    MEASURE.  197 

3.  How  many  cubic  feet  in  a  stick  of  timber  whose 
larger  end  is  25  feet  by  20,  the  smaller  15  feet  by  10, 
and  the  length  12  feet? 

Arts.  3700. 

4.  What  is  the  number  of  cubic  feet  in  a  stick  of  hewn 
timber,  whose  ends  are  30  inches  by  27  and  24  inches 
by  18,  the  length  being  24  feet? 

Ans.  102. 

«5.  The  length  of  a  piece  of  timber  is  20.38  feet,  and  the 
ends  are  unequal  squares :  the  side  of  the  greater  is  19£ 
inches,  and  of  the  less  9£  inches :  what  is  the  solid  content  ? 

Ans.  30.763  cubic  ft. 

6.  The  length  of  a  piece  of  timber  is  27.36  feet:  at 
the  greater  end,  the  breadth  is  1.78  feet  and  the  thickness 
1.23  feet;  and  at  the  less  end,  the  breadth  is  1.04  feet 
and  the  thickness  0.91  feet:   what  is  its  solidity? 

Ans.  41.8179  cubic  ft. 

8.  How  do  you  do  when  the  timber  does  not  taper  regu- 
larly  ? 

If  the  timber  does  not  taper  regularly,  measure  parts  of 
the  stick,  the  same  as  if  it  had  a  regular  taper,  and  take 
the  sum  of  the  parts  for  the  entire  solidity. 


9.  Knowing  the  area  of  the  end  of  a  square  piece  of  tim- 
ber which  does  not  taper,  how  do  you  find  the  length  which 
must  be  cut  off  in  order  to  obtain  a  given  solidity  ? 

1st.  Reduce  the  given  solidity  to  cubic  inches. 

2d.  Divide  the  number  of  cubic  inches  by  the  area  of 
the  end  expressed  in  inches,  and  the  quotient  will  be  the 
length  in  inches. 


198  BOOK    VI. SECTION    II. 

EXAMPLES. 

1.  A  piece  of  timber  is  10  inches  square  :  how  much 
must  be  cut  off  to  make  a  solid  foot  ? 

10  x  10  =  100  square  inches. 
Then,  1728  h-  100  =  17.28  inches. 

2.  A  piece  of  timber  is  20  inches  broad  and  10  inches 
deep :   how  much  in  length  will  make  a  solid  foot  ? 

Ans.  8|£  in. 

3.  A  piece  of  timber  is  9  inches  broad  and  6  inches 
deep:   how  much  in  length  will  make  3  solid  feet? 

Ans.  8  ft. 

10.  How  do  you  find  the  solidity  of  round  or  unsquarcd 
timber  ?  W 

1st.  Take  the  girt  or  circumference,  and  then  divide  it 
by  5. 

2d.  Multiply  the  square  of  one-fifth  of  the  girt  by  twice 
the  length,  and  the  product  will  be  the  solidity  very  nearly. 

EXAMPLES. 

1.  A  piece  of  round  timber  is  9f  feet  in  length,  and  the 
girt  is  13  feet:  what  is  its  solidity? 

First,  13  -f-  5  =  2.6  the  fifth  of  the  girt. 
Also,  2l?  =  6.76  ;  and  9.75  X  2  =  19.50. 
Again,       6.76  x  19.5  =  131.82  cubic  feet,  which  is  the 
required  solidity. 

2.  The  length  of  a  tree  is  24  feet,  and  the  girt  through- 
out 8  feet:   what  is  the  content? 

Ans.  122.88  cubic  ft. 

3.  Required  the  content  of  a  piece  of  timber,  its  length 
being  9  feet  6  inches,  and  girt  14  feet. 

Ans.  148.96  cubic  ft. 


OF    LOGS    FOR    SAWING.  199 

11.  How  do  you  do  when  the  timber  tapers? 

Gird  the  timber  at  as  many  points  as  may  be  neces- 
sary, and  divide  the  sum  of  the  girts  by  their  number  for 
the  mean  girt,  of  which  take  one-fifth,  and  proceed  as 
before . 

4.  If  a  tree,  girt   14  feet  at  the  thicker  end  and  2  feet 

at  the  smaller  end,  be  24  feet  in  length,  how  many  solid 

feet  will  it  contain? 

Ans.  122.88. 

5.  A  tree  girts  at  five  different  places  as  follows :  in 
the  first  9.43  feet;  in  the  second  7.92  feet;  in  the  third 
6.15  feet;  in  the  fourth  4.74  feet;  and  in  the  fifth  3.16 
feet:  now,  if  the  length  of  the  tree  be  17.25  feet,  what  is 
its  solidity? 

Ans.  54.42499  cubic  ft. 


OF    LOGS    FOR    SAWING. 

12.  What  is  often  necessary  for  lumber  merchants  ? 

It  is  often  necessary  for  lumber  merchants  to  ascertain 
the  number  of  feet  of  boards  which  can  be  cut  from  a 
given  log ;  or,  in  other  words,  to  find  how  many  logs  will 
be  necessary  to  make  a  given  amount  of  boards. 

13.  What  is  a  standard  board? 

A  standard  board  is  one  which  is  12  inches  wide,  1 
inch  thick,  and  12  feet  long:  hence,  a  standard  board  is 
1  inch  thick  and  contains  12  square  feet. 

14.  What  is  a  standard  saw-log? 

A  standard  log  is  12  feet  long  and  24  inches  in  diam- 
eter. 


200  BOOK    VI. SECTION    II. 

15.  How  will  you  find  the  number  of  feet  of  boards  which 
can  be  sawed  from  a  standard  log  1 

If  we    saw  off,  say  2   inches,  from  each  side,  the  log 
will  be  reduced  to  a  square  20  inches  on  a  side.     Now, 
since  a  standard  board  is  one  inch  in  thickness,  and  since 
the  saw  cuts   about  one  quarter  of  an  inch  each  time  it 
goes  through,  it  follows  that  one-fourth  of  the  log  will  be 
consumed  by  the  saw.     Hence  we  shall  have 
3 
20  X  -—  =  the  number  of  boards  cut  from  the  log. 
4 

Now,  if  the  width  of  a  board  in  inches  be  divided  by  12, 

and  the  quotient  be   multiplied  by  the   length  in  feet,  the 

product  will  be  the  number  of  square   feet  in   the   board. 

20 
Hence,    —  x  length  of  the  log  in  feet  =  the  square  feet 
12 

in  each  board.     Therefore, 

20  X  4- 

4 
the  boards, 

3  2  1 

=  20  X  1 .0  X  —  X  —  X  length  of  log  =  20  X  10  X  —  X 

4  12  o 

length;    and  the   same   may  be  shown   for  a    log  of  any 
length. 

1 6.  What  then  is  the  rule  for  finding  the  number  of  feet 
of  boards  which  can  be  cut  from  any  log  whatever  ? 

From  the  diameter  of  the  log,  in  inches,  subtract  4  for 
the  slabs.  Then  multiply  the  remainder  by  half  itself  and 
the  product  by  the  length  of  the  log,  in  feet,  and  divide  the 
result  by  8  :  the  quotient  will  be  the  number  of  square  feet 

EXAMPLES. 

1.  What  is  the  number  of  feet  of  boards  which  can  be 
cut  from  a  standard  log? 


3        20 
20  x  —  X  —  X  length  of  log  ==  the   square  feet  in  all 


bricklayers'  work.  201 


Diameter 
for  slabs 
remainder 
half  remainder 

24  inches 

4 
20 
10 

200 

length  of  log          12 

8)2400 

300  =  the  number  of  feet. 

2.  How  many  feet  can  be  cut  from  a  log  12  inches  in 
diameter  and  12  feet  long?  .         ._ 


3.  How  many  feet  can  be  cut  from  a  log  20  inches  in 

Ans.  256. 


diameter  and  16  feet  long? 


4.  How  many  feet  can  be  cut  from  a  log  24  inches  in 

diameter  and  16  feet  long?  .         .__ 

6  Ans.  400. 

5.  How  many  feet  can  be  cut  from  a  log  28  inches  in 
diameter  and  14  feet  long?  .        __. 


SECTION   III. 


BRICKLAYERS     WORK. 


1.  In  how  many  ways  is  artificers1  work  computed? 

Artificers'  work,  in  general,  is  computed  by  three  differ- 
ent measures,  viz. : 

1st.  The  linear  measure,  or,  as  it  is  called  by  mechanics, 
running  measure. 

2d.  Superficial  or  square  measure,  in  which  the  compu- 
tation is  made  by  the  square  foot,  square  yard,  or  by  the 
square  containing  100  square  feet,  or  yards. 

9* 


202  BOOK    VI. SECTION    III. 

3d.  By  the  cubic  or  solid  measure,  when  it  is  estimated 
by  the  cubic  foot,  or  the  cubic  yard.  The  work,  however, 
is  often  estimated  in  square  measure,  and  the  materials  for 
construction  in  cubic  measure. 

2.  What  proportion  do  the  dimensions  of  a  brick  bear  to 
each  other? 

The  dimensions  of  a  brick  generally  bear  the  following 

proportions  to  each  other,  viz. : 

Length  =  twice  the  width,  and 
Width   =  twice  the  thickness,  and 

hence,  the  length  is  equal  to  four  times  the  thickness. 

3.  What  are  the  common  dimensions  of  a  brick  ?  How 
many  cubic  inches  does  it  contain  ? 

The  common  length  of  a  brick  is  8  inches,  in  which 
case  the  width  is  4  inches,  and  the  thickness  2  inches. 
A  brick  of  this  size  contains 

8  X  4  X  2  =  64  cubic  inches  ;  and  since  a  cubic  foot 
contains  1728  cubic  inches,  we  have 

1728  -i-  64  =  27  the  number  of  bricks  in  a  cubic  foot. 

4.  If  a  brick  is  9  inches  long,  what  will  be  its  width  and 
what  its  content  ? 

If  the  brick  is  9  inches  long,  then  the  .width  is  4£  inches, 
and  the  thickness  2\  ;  and  then  each  brick  will  contain 

9  X  4|  x  2}  =  91|   cubic  inches  in  each  brick;  and 
1728  —  91  i  =  19  nearly,  the  number  of  bricks  in  a  cubic 

foot.     In  the  examples  which  follow,  we  shall  suppose  the 
brick  to  be  8  inches  long. 

5.  How  do  you  find  the  number  of  bricks  required  to  build 
0   wall  of  given  dimensions  ? 

let.  Find  the  content  of  the  wall  in  cubic  feet. 


bricklayers'  work.  203 

2d.  Multiply  the  number  of  cubic  feet  by  the  number  of 
bricks  in  a  cubic  foot,  and  the  result  will  be  the  number 
of  bricks  required. 

EXAMPLES. 

1.  How  many  bricks,  of  8  inches  in  length,  will  be  re- 
quired to  build  a  wall  30  feet  long,  a  brick  and  a  half 

thick,  and  15  feet  in  height? 

Ans.  12150. 

2.  How  many  bricks,  of  the  usual  size,  will  be  required 
to  build  a  wall  50  feet  long,  2  bricks  thick,  and  36  feet 

in  heiSht-  -  .  Ans.  64800. 

6.    What  allowance  is  made  for  the  thickness  of  the  mortar  ? 

The  thickness  of  mortar  between  the  courses  is  nearly 
a  quarter  of  an  inch,  so  that  four  courses  will  give  nearly 
1  inch  in  height.  The  mortar,  therefore,  adds  nearly 
one-eighth  to  the  height;  but  as  one-eighth  is  rather  too 
large  an  allowance,  we  need  not  consider  the  mortar  which 
goes  to  increase  the  length  of  the  wall. 

3.  How  many  bricks  would  be  required  in  the  first  and 
second  examples,  if  we  make  the  proper  allowance  for 
mortar  ? 

Ans.   \  lst'  10631*' 


<  lst. 
<2d. 


56700. 


7.  How  do  bricklayers  generally  estimate  their  work? 

Bricklayers  generally  estimate  their  work  at  so  much 
per  thousand  bricks.  To  find  the  value  of  things  estimated 
by  the  thousand,  see  Arithmetic,  page  192. 

4.  What  is  the  cost  of  a  wall  60  feet  long,  20  feet  high, 
and  two  and  a  half  bricks  thick,  at  $7.50  per  thousand, 
which  price  we  suppose  to  include  the  cost  of  the  mortar ? 


204  BOOK    VI.— SECTION    III 

If  we  suppose  the  mortar  to  occupy  a  space  equal  to 
one-eighth  the  height  of  the  wall,  we  must  find  the  quantity 
of  bricks  under  the  supposition  that  the  wall  was  17^  feet 

in  hei§ht-  Arts.  $354,371. 

8.  In  estimating  the  bricks  for  a  house,  .what  allowances 
are  made? 

In  estimating  the  bricks  for  a  house,  allowance  must  be 
made  for  the  windows  and  doors. 


OF    CISTERNS* 

9.  What  are  cisterns? 

Cisterns  are  large  reservoirs  constructed  to  hold  watei, 
and  to  be  permanent,  should  be  made  either  of  brick  or 
masonry. 

It  frequently  occurs  that  they  are  to  be  so  constructed 
as  to  hold  given  quantities  of  water,  and  it  then  becomes 
a  useful  and  practical  problem  to  calculate  their  exact 
dimensions. 

10.  How  many  cubic  inches  in  a  hogshead? 

It  was  remarked  in  Arithmetic,  page  104,  that  a  hogs- 
head contains  63  gallons,  and  that  a  gallon  contains  231 
cubic  inches.  Hence,  231  X  63  =  14553,  the  number  of 
cubic  inches  in  a  hogshead. 

11.  How  do  you  find  the  number  of  hogsheads  which  a 
cistern  of  given  dimensions  will  contain? 

1st.  Find  the  solid  content  of  the  cistern  in  cubic  inches. 
2d.  Divide  the  content  so  found  by  14553,  and  the  quo* 
dent  will  be  the  number  of  hogsheads. 


OF    CISTERNS.  205 

EXAMPLE. 

The  diameter  of  a  cistern  is  6  feet  6  inches,  and  height 
lO  feet:  how  many  hogsheads  does  it  contain? 

The  dimensions  reduced  to  inches  are  78  and  120.  To 
find  the  solid  content,  see  page  162.  Then,  the  content 
in  cubic  inches,  which  is  573404.832,  gives 

573404.832  -r-  14553  =  39.40  hogsheads,  nearly. 

12.  If  the  height  of  a  cistern  be  given,  how  do  you  find 
the  diameter,  so  that  the  cistern  shall  contain  a  given  number 
of  hogsheads  ? 

1st.  Reduce  the  height  of  the  cistern  to  inches,  and  the 
content  to  cubic  inches. 

2d.  Multiply  the  height  by  the  decimal  .7854. 

3d.  Divide  the  content  by  the  last  result,  and  extract 
the  square  root  of  the  quotient,  which  will  be  the  diameter 
of  the  cistern  in  inches. 

EXAMPLE. 

The  height  of  a  cistern  is  10  feet:  what  must  be  its 
diameter,  that  it  may  contain  40  hogsheads  ? 

Ans.  78.6  in.  nearly. 

13.  If  the  diameter  of  a  cistern  be  given,  how  do  you  find 
the  height,  so  that  the  cistern  shall  contain  a  given  number 
of  hogsheads  ? 

1st.  Reduce  the  content  to  cubic  inches. 
2d.  Reduce  the  diameter  to  inches,  and  then  multiply  its 
square  by  the  decimal  .7854. 

3d.  Divide  the  content  by  the  last  result,  and  the  quo- 
tient will  be  the  height  in  inches. 


206  BOOK    VI. SECTION    IV 

EXAMPLE. 

The  diameter  of  a   cistern  is  8  feet :  what  must  be  its 
height  that  it  may  contain  150  hogsheads? 

Ans.  25  ft.  1  in.  nearly. 


SECTION   IV. 

masons'  work. 


1.  What  belongs  to  masonry,  and  what  measures  are  used? 

All  sorts  of  stone  work.  The  measure  made  use  of  is 
either  superficial  or  solid. 

Walls,  columns,  blocks  of  stone  or  marble,  are  measured 
by  the  cubic  foot ;  and  pavements,  slabs,  chimney-pieces, 
&c,  are  measured  by  the  square  or  superficial  foot.  Cubic 
or  solid  measure  is  always  used  for  the  materials,  and  the 
square  measure  is  sometimes  used  for  the  workmanship. 

EXAMPLES. 

1.  Required  the  solid  content  of  a  wall  53  feet  6  inches 
long,  12  feet  3  inches  high,  and  2  feet  thick. 

Ans.  1310f  ft. 

2.  What  is  the   solid   content  of  a  wall,  the  length  of 

which  is  24  feet  3  inches,  height   10  feet  9  inches,   and 

thickness  2  feet?  ,        __.  nry_    ~ 

Ans.  521.375  ft. 

3.  In  a  chimney-piece  we  find  the  following  dimensions : 
Length  of  the  mantel  and  slab,     4  feet    2  inches. 
Breadth  of  both  together, 
Length  of  each  jamb, 
Breadth  of  both, 
Required  the  superficial  content. 


3 

"      2 

tt 

4 

"      4 

tt 

1 

"       9 

« 

Ans. 

21 

ft.  10'. 

carpenters'  and  joiners'  work.  207 

SECTION  V. 
carpenters'  and  joiners'  work. 

1.  In  what  does  carpenters'  and  joiners'1  work  consist  ? 
Carpenters'  and  joiners'  work  is  that  of  flooring,  roofing 

<fcc.   and  is  generally  measured  by  the  square  of  100  square 
feet. 

2.  When  is  a  roof  said  to  have  a  true  pitch  ? 

In  carpentry,  a  roof  is  said  to  have  a  true  pitch  when 
the  length  of  the  rafters  is  three-fourths  the  breadth  of  the 
building.  The  rafters  then  are  nearly  at  right  angles.  It 
is  therefore  customary  to  take  once  and  a  half  times  the 
area  of  the  flat  of  the  building  for  the  area  of  the  roof. 

EXAMPLES. 

1.  How  many  squares,  of  100  square  feet  each,  in  a 
floor  48  feet  6  inches  long,  and  24  feet  3  inches  broad? 

Ans.   11  and  76}  sq.ft. 

2.  A  floor  is  36  feet  3  inches  long,  and  16  feet  6  inches 
broad:  how  many  squares  does  it  contain? 

Ans.  5  and  98}  sq.  ft. 

3.  How  many  squares  are  there  in  a  partition  91  feet 
9  inches  long,  and  11  feet  3  inches  high? 

Ans.  10  and  32  sq.ft. 

4.  If  a  house  measure  within  the  walls  52  feet  8  inches 
in  length,  and  30  feet  6  inches  in  breadth,  and  the  roof  be 
of  the  true  pitch,  what  ~vill  the  roofing  cost  at  $1.40  per 

6qUare?  Ans.  $33,733. 


208  BOOK    VI. SECTION    V. 

*OF    BINS    FOR    GRAIN. 

3.  What  is  a  bin? 

It  is  a  wooden  box  used  by  farmers  for  the  storage  of 
their  grain. 

4.  Of  what  form  are  bins  generally  made  1 

Their  bottoms  or  bases  are  generally  rectangles,  and 
horizontal,  and  their  sides  vertical. 

5.  How  many  cubic  feet  are  there  in  a  bushel  ? 

Since  a  bushel  contains  2150.4  cubic  inches,  (see  Arith- 
metic, page  106,)  and  a  cubic  foot  1728  inches,  it  follows 
that  a  bushel  contains  one  and  a  quarter  cubic  feet,  nearly. 

6.  Having  any  number  of  bushels,  how  then  will  you  find 
the  corresponding  number  of  cubic  feet  ? 

Increase  the  number  of  bushels  one-fourth  itself,  and  the 
result  will  be  the  number  of  cubic  feet. 

EXAMPLES. 

1 .  A  bin  contains  372  bushels ;  how  many  cubic  feet 
does  it  contain? 

372  -L  4  =  93  ;  hence,  372  -f  93  =  465  cubic  feet. 

2.  In  a  bin  containing  400  bushels,  how  many  cubic 
feet?  Ans.  500. 

7.  How  will  you  find  the  number  of  bushels  which  a  bin 
of  a  given  size  will  hold  ? 

Find  the  content  of  the  bin  in  cubic  feet ;  then  diminish 
the  content  by  one-fifth,  and,  the  result  will  be  the  content 
in  bushels. 

3.  A  bin  is  8  feet  long,  4  feet  wide,  and  5  feet  high  • 
how  many  bushels  will  it  hold? 


OF    BINS    FOR    GRAIN.  209 

8x4x5  =  160 
then,     160  -4-  5  =    32  :   160  —  32  =  128  bushels  = 
capacity  of  bin. 

4.  How  many  bushels  will  a  bin  contain  which  is  7  feet 
long    3  feet  wide,  and  6  feet  in  height? 

Ans.  100.8  bush. 

8.  How  will  you  find  the  dimensions  of  a  bin  which  shall 
contain  a  given  number  of  bushels : 

Increase  the  number  of  bushels  one-fourth  itself,  and  the 
result  will  show  the  number  of  cubic  feet  which  the  bin 
will  contain.  Then,  when  two  dimensions  of  the  bin  are 
known,  divide  the  last  result  by  their  product,  and  the  quo- 
tient will  be  the  other  dimension. 

5.  What  must  be  the  height  of  a  bin  that  will  contain 
600  bushels,  its  length  being  8  feet  and  breadth  4  ? 

600  -r-  4  =  150  ;  hence,  600  +  150  =  750  =  the  cubic  feet , 
and  8x4  =  32,  the  product  of  the  given  dimensions. 
Then,  750  -»  32  =  23.44  feet,  the  height  of  the  bin. 

6.  What  must  be  the  width  of  a  bin  that  shall  contain 
900  bushels,  the  height  being  12  and  the  length  10  feet? 
900  -4-  4  =  225  ;  hence,  900  +  225  =  1125  =  the  cubic  feet ; 
and  12  x  10  =  120,  the  product  of  the  given  dimensions. 
Then,  1125  -4-  120  =  9.375  feet,  the  width  of  the  bin. 

7.  The  length  of  a  bin  is  4  feet,  its   breadth  5  feet  6 

inches:   what  must  be  its  height  that  it  may  contain  136 

bushels  ?  A        „   /.    Q  . 

Ans.  7  ft.  8«2;i.  -f 

8.  The  depth  of  a  bin  is  6  feet  2  inches,  the  breadth 
4  feet  8  inches :  what  must  be  the  length  that  'it  may  con- 
tain 200  bushels?.  ,        1Ay. 

Ans.  104  in.  -f* 


210  BOOK    VI. — SECTION   VII. 

SECTION   VI. 

SLATERS'    AND    TILERS*    WORK. 

1.  How  is  the  content  of  a  roof  found? 

In  this  work,  the  content  of  the  roof  is  found  by  multi- 
plying the  length  of  the  ridge  by  the  girt  from  eaves  to 
eaves.  Allowances,  however,  must  be  made  for  the  double 
rows  of  slate  at  the  bottom. 

EXAMPLES. 

1.  The  length  of  a  slated  roof  is  45  feet  9  inches,  and 
its  girt  34  feet  3  inches :  what  is  its  content  1 

Ans.  1566.9375  sq.ft. 

2.  What  will  tne  tiling  of  a  barn  cost,  at  $3.40  per 
square  of  100  feet,  the  length  being  43  feet  10  inches, 
and  breadth  27  feet  5  inches,  on  the  flat,  the  eave-board 
projecting  16  inches  on  each  side,  and  the  roof  being  of 
the  true  pitch? 

Ans.  $65.26. 


SECTION   VII. 

plasterers'  work. 


1.  How  many  kinds  of  plasterers1  work  are  there,  and  how 
are  they  measured? 

Plasterers'  work  is  of  two  kinds,  viz. :  ceiling,  which  is 
plastering  on  laths ;  and  rendering,  which  is  plastering  on 
walls.     These  are  measured  separately. 


plasterers'  work.  211 

The  contents  are  estimated  either  by  the  square  foot, 
the  square  yard,  or  by  the  square  of  100  feet. 

Inriched  mouldings,  &c,  are  rated  by  the  running  or 
lineal  measure. 

In  estimating  plastering,  deductions  are  made  for  chim- 
neys, doors,  windows,  &c. 

EXAMPLES. 

1.  How  many  square  yards  are  contained  in  a  ceiling 
43  feet  3  inches  long,  and  25  feet  6  inches  broad? 

Ans.   122^  nearly 

2.  What  is  the  cost  of  ceiling  a  room  21  feet  8  inches, 
by  14  feet  10  inches,  at  18  cents  per  square  yard? 

Ans.  $6,421. 

3.  The  length  of  a  room  is  14  feet  5  inches,  breadth 
13  feet  2  inches,  and  height  to  the  under  side  of  the  cor- 
nice 9  foet  3  inches.  The  cornice  girts  8i  inches,  and 
projects  5  inches  from  the  wall  on  the  upper  part  next  the 
ceiling,  deducting  only  for  one  door  7  feet  by  4  :  what  will 
be  the  amount  of  the  plastering? 

r  53  yds.  5  ft.  3/  6"  of  rendering. 
Ans.    <  18  yds.  5  ft.  6/  4"  of  ceiling. 
{37  ft.  10'  9"  of  cornice. 
How  is  the  area  of  the  cornice  found  in   the  above  exam- 
files  ? 

The  mean  length  of  the  cornice,  both  in  the  length  and 
breadth  of  the  house,  is  found  by  taking  the  middle  line  of 
the  cornice.  Now,  since  the  cornice  projects  5  inches  at 
the  ceiling,  it  will  project  2\  inches  at  the  middle  line  ; 
and  therefore,  the  length  of  the  middle  line  along  the  length 
of  the  room  will  be  14  feet,  and  across  the  room,  12  feet 
9  inches.  Then  multiply  the  double  of  each  of  these  num- 
bers by  the  girth,  which  is  8l  inches,  and  the  sum  of  the 
products  will  be  the  area  of  the  cornice. 


212  BOOK    VI. — SECTION    IX., 

SECTION  VIII. 

PAINTERS'    WORK. 

How  is  painters'  work  computed,  and  what  allowances  are 
made  ? 

Painters'  work  is  computed  in  square  yards.  Every  part 
is  measured  where  the  color  lies,  and  the  measuring  line 
is  carried  into  all  the  mouldings  and  cornices. 

Windows  are  generally  done  at  so  much  a  piece.  It  is 
usual  to  allow  double  measure  for  carved  mouldings,  &c. 

EXAMPLES. 

1.  How  many  yards  of  painting  in  a  room  which  is  65 
feet  6  inches  in  perimeter,  and  12  feet  4  inches  in  height  ? 

Ans.  89f  £  sq.  yds. 

2.  The  length  of  a  room  is  20  feet,  its  breadth  14  feet 
6  inches,  and  height  10  feet  4  inches  :  how  many  yards 
of  painting  are  in  it,  deducting  a  fire-place  of  4  feet  by 
4  feet  4  inches,  and  two  windows,  each  6  feet  by  3  feet 
2  inches?  Ans.  73^  sq.  yd*. 


SECTION   IX. 

pavers'  work. 


How  is  pavers1  work  estimated? 

Pavers'  work  is  done  by  the  square  yard,  and  the  con  ' 
tent  is  found   by  multiplying   the    length   and  breadth  to 
gether. 

EXAMPLES. 

1.  What  is  the   cost  of  paving  a  side-walk,  the  length 


PLUMBERS     WORK. 


213 


of  which  is  35  feet  4  inches,  and  breadth  8  feet  3  inches, 
at  54  cents  per  square  yard?  Ang    $lJAQ  Q 

2.  What  will  be  the  cost  of  paving  a  rectangular  court- 
yard, whose  length  is  63  feet,  and  breadth  45  feet,  at  2a*. 
6d.  per  square  yard ;  there  being,  however,  a  walk  running 
lengthwise  5  feet  3  inches  broad,  which  is  to  be  flagged 
with  stone  costing  3  shillings  per  square  yard? 

Ans.  £40  5s.  \Q\d. 


SECTION   X. 


PLUMBERS     WORK. 

1.  Plumbers'  work  is  rated  at  so  much  a  pound,  or  else 
by  the  hundred  weight.  Sheet  lead,  used  for  gutters,  &c, 
weighs  from  6  to  12  lbs.  per  square  foot.  Leaden  pipes 
vary  in  weight  according  to  the  diameter  of  their  bore  and 
thickness. 

The  following  table  shows  the  weight  of  a  square  foot 
of  sheet  lead,  according  to  its  thickness;  and  the  common 
weight  of  a  yard  of  leaden  pipe,  according  to  the  diameter 
of  the  bore. 


Thickness 
of  lead. 

Pounds  to  a  square 
foot. 

Bore  of 
leaden  pipes. 

Pounds 
per  yard. 

Inch. 
TV 

5.899 

Inch. 

Of 

10 

* 

6.554 

1 

12 

1 

8 

7.373 

H 

16 

♦ 

8.427 

H 

18 

1 

9.831 

if 

21 

i 

11.797 

2 

24 

214  BOOK    VI. — SECTION   X. 

EXAMPLES. 

1.  What  weight  of  lead  of  T\  of  an  inch  in  thickness, 
will  cover  a  flat  15  feet  6  inches  long,  and  10  feet  3  inches 
broad,  estimating  the  weight  at  6  lbs.  per  square  foot? 

Ans.  8  cwt.  2  qr.  l±lb. 

2.  What  will  be  the  cost  of  130  yards  of  leaden  pipe 
of  an  inch  and  a  half  bore,  at  8  cents  per  pound,  sup- 
posing each  yard  to  weigh  18  lbs.? 

Ans.  $187.20. 

3.  The  lead  used  for  a  gutter  is  12  feet  5  inches  long 
and  1  foot  3  inches  broad :  what  is  its  weight,  supposing  it 
to  be  i  of  an  inch  in  thickness? 

Ans.  101  lbs.  12  oz.  13.6  dr. 

4.  What  is  the  weight  of  96  yards  of  leaden  pipe,  of 
an  inch  and  a  quarter  bore? 

Ans.  13  cwt.  2  qr.  24  lbs. 

5.  What  will  be  the  cost  of  a  sheet  of  lead  16  feet  6 
inches  long  and  10  feet  4  inches  broad,  at  5  cenfs  per 
pound;  the  lead  being  |  of  an  inch  in  thickness? 

Ans.  883.81. 


OP    MATTER    AND    BODIES.  215 


BOOK  VII 


INTRODUCTION    TO    MECHANICS. 
SECTION   I. 

OF    MATTER    AND    BODIES. 

1.  What  is  matter? 

Matter  is  a  general  name  for  every  thing  which  has 
substance,  and  is  always  capable  of  being  increased  or  di- 
minished. Whatever  we  can"  touch,  taste,  smell,  or  see,  is 
matter. 

2.  What  is  a  body? 

A  body  is  any  portion  of  matter. 

3.  What  is  space  ?     How  many  dimensions  has  it  ? 

Space  is  mere  extension,  in  which  all  bodies  are  sit- 
uated. Thus,  when  a  body  has  a  certain  place,  it  is  said 
to  occupy  that  portion  of  space  which  it  fills.  Space  has 
three  dimensions — length,  breadth,  and  thickness. 

4.  What  are  the  properties  common  to  all  bodies? 

The  properties  which  belong  to  all  bodies  are :  impene- 
trability, extension,  figure,  divisibility,  inertia,  and  attraction. 

5.  What  is  impenetrability? 

Impenetrability  is  the  property,  in  virtue  of  which  a 


216  BOOK    VII. SECTION    I. 

body  must  fill  a  certain  space,  and  which  no  other  body 
can  occupy  at  the  same  time.  Thus,  if  you  fill  a  vessel 
full  of  water,  and  then  plunge  in  your  hand,  or  a  stick, 
some  of  the  water  will  be  forced  over  the  top  of  the  ves- 
sel. Your  hand  or  the  stick  removes  the  water,  and  does 
not  occupy  the  space  until  after  the  water  is  displaced. 

6.  What  is  extension? 

Since  a  body  occupies  space,  it  must,  like  any  portion 
of  space,  have  the  dimensions  of  length,  breadth,  and  thick- 
ness. These  are  called  the  dimensions  of  extension,  and 
vary  in  different  bodies.  The  length,  breadth,  and  depth 
of  a  house  are  very  different  from  those  of  an  inkstand. 

7.  How  are  length  and  breadth  measured?  How  do  you 
measure  height  and  depth? 

Length  and  breadth  are  generally  measured  in  a  hori- 
zontal direction.  Height  and  depth  are  the  same  dimen- 
sion :  height  is  measured  upward,  and  depth  downward. 
Thus,  we  say  a  mountain  is  400  feet  high,  and  a  river  50 
feet  deep. 

8.  What  is  figure  ? 

Figure  is  merely  the  limit  of  extension.  Figure  is  also 
called  form  or  shape. 

9.  When  is  a  body  said  to  be  regular?    when  irregular? 
If  all  the  parts  of  a  body  are  arranged  in  the  same  way, 

about  a  line  or  a  centre,  the  body  is  said  to  be  regular  or 
symmetrical ;  and  when  the  parts  are  not  so  arranged,  the 
body  is  said  to  be  irregular.  Nature  has  given  regular 
forms  to  nearly  all  her  productions. 

10.  What  is  divisibility  ? 

Divisibility   denotes  the  susceptibility  of  matter  to   be 


OF    MATTER    AND    BODIES.  217 

continually  divided.  That  is,  a  portion  of  matter  may  be 
divided,  and  each  part  again  divided,  and  each  of  the  parts 
divided  again,  and  so  on,  continually,  without  ever  arriving 
at  a  portion  which  will  be  absolutely  nothing. 

Suppose,  for  instance,  you  take  a  portion  of  matter,  say 
one  pound  or  one  ounce,  and  divide  it  into  two  equal  parts, 
and  then  divide  each  part  again  into  two  equal  parts,  and 
so  on  continually.  Now,  all  the  parts  will  continually  grow 
smaller  and  smaller,  but  no  one  of  them  will  ever  become 
equal  to  nothing,  since  the  half  of  a  thing  must  always 
have  some  value. 

11.  What  is  inertia? 

Inertia  is  the  resistance  which  matter  makes  to  a  change 
of  state.  Bodies  are  not  only  incapable  of  changing  their 
actual  state,  whether  it  be  that  of  motion  or  rest,  but  they 
seem  endowed  with  the  power  of  resisting  such  a  change. 
This  property  is  called  inertia. 

12.  If  a  body  is  at  rest,  will  it  remain  so  ?  If  in  mo- 
tion, will  it  continue  so  ? 

If  a  body  is  at  rest  it  will  remain  so,  unless  something 
be  applied  from  without  to  move  it ;  and  if  it  be  moving, 
it  will  continue  to  move,  unless  something  stops  it. 

13.  What  are  atoms  ?  » 

The  smallest  parts  into  which  we  can  suppose  a  body 
divided,  are  called  particles  or  atoms. 

14.  Do  these  atoms  adhere  to  each  other? 
They  do,  and  form  masses  or  bodies. 

15.  What  is  the  force  called  which  unites  them? 

If  is  called  the  attraction  of  cohesion.  Without  this  power 
solid  bodies  would  crumble  to  pieces  and  fall  to  atoms. 

10 


218  BOOK    VII. SECTION    I. 

16.  In  what  kind  of  bodies  does  this  attraction  exist? 
In  all  bodies,  fluid  as  well  as  solid.     It  is  the  attraction 

of  cohesion  which  holds  a  drop  of  water  in  suspension  at 
the  end  of  the  finger,  and  causes  it  to  take  a  spherical 
form. 

The  attraction  of  cohesion  is  stronger  in  some  substances 
than  in  others.  Those  in  which  it  is  the  weakest  are  easily 
broken,  or  the  attraction  is  easily  overcome ;  while  those 
in  which  it  is  greater,  are  proportionably  stronger. 

17.  What  is  the  difference  between  the  attraction  of  cohe- 
sion and  the  attraction  of  gravitation  ? 

The  attraction  of  cohesion  unites  the  particles  of  matter, 
and  these  by  their  aggregation  form  masses  or  bodies.  The 
attraction  of  gravitation  is  the  force  by  which  masses  of 
matter  tend  to  come  together.  The  attraction  of  cohesion 
acts  only  between  particles  of  matter  which  are  very  near 
each  other,  while  the  attraction  of  gravitation  acts  between 
bodies  widely  separated. 

18.  Is  the  attraction  between  bodies  mutual? 

The  attraction  between  two  bodies  is  mutual;  that  is, 
each  body  attracts  the  other  just  as  much,  and  no  more, 
than  it  is  attracted  by  it.  But  if  the  bodies  are  left  free, 
the  smaller  will  move  towards  the  larger ;  for,  as  they  are 
urged  together  by  equal  forces,  the  smaller  will  obey  the 
force  faster  than  the  larger.  Thus,  the  earth  being  larger 
than  any  body  near  its  surface,  forces  all  bodies  towards 
it,  and  they  immediately  fall  unless  the  attraction  of  gravi- 
tation is  counteracted. 

It  should,  however,  be  borne  in  mind  that  every  body 
altracts  the  earth  just  as  strongly  as  the  earth  attracts  the 
body  ;  and  the  body  moves  towards  the  earth,  only  because 
the  ea  rth  is  larger,  and  therefore  not  as  rapidly  moved  by 
their  mutual  attraction 


LAWS    OF    MOTION,    ETC.  219 

19.    What  is  weight? 

Weight  is  the  force  which  is  necessary  to  overcome  the 
attraction  of  gravitation.  Thus,* if  we  have  two  bodies, 
and  one  has  twice  as  much  tendency  to  descend  towards 
the  earth  as  the  other,  it  will  require  just  twice  as  much 
force  to  support  it,  and  hence  we  say  that  it  is  twice  as 
heavy. 


SECTION   II. 


LAWS  OF  MOTION,  AND  CENTRE  OF  GRAVITY. 

1.  What  is  motion? 

Motion  is  a  change  of  place.  Thus,  a  body  is  said  to 
be 'in  motion  when  it  is  continually  changing  its  place. 

2.  Can  a  body  put  in  motion  stop  itself? 

It  has  been  observed  in  Art.  11,  that  bodies  are  indiffer- 
ent to  rest  or  motion.  Hence,  a  body  cannot  put  itself  in 
motion,  or  stop  itself  after  it  has  begun  to  move. 

3.  What  is  force  or  power  ? 

That  which  puts  a  body  in  motion,  or  which  changes 
its  motion  after  it  has  begun  to  move,  is  called  force  or 
power.  Thus,  the  stroke  of  the  hammer  is  the  force  which 
drives  the  nail,  the  effort  of  the  horse  the  force  which 
moves  the  carriage,  and  the  attraction  of  gravitation  the 
force  which  draws  bodies  to  the  earth. 

4.  What  is  velocity  ? 

The  rate  at  which  a  body  moves,  or  the  rapidity  ot  its 
motion,  is  estimated  bv  the  space  which  it  passes  over  in 


220  BOOK    VII. SECTION    II. 

a  given  portion  of  time,  and  this  rate  is  called  its  velo- 
city.  Thus,  if  in  one  minute  of  time  a  body  passes  over 
200  feet,  its  velocity  is  said  to  be  200  feet  per  minute ; 
and  if  another  body,  in  the  same  time,  passes  over  400 
feet,  its  velocity  is  said  to  be  400  feet  per  minute,  or 
double  that  of  the  first. 

5.  What  is  uniform  velocity? 

When  a  body  moves  over  equal  distances  in  equal  times, 
its  velocity  is  said  to  be  uniform.  Thus,  if  a  body  move 
at  the  rate  of  30  feet  a  second,  it  has  a  uniform  velocity, 
for  it  always  passes  over  an  equal  space  in  an  equal  time. 

6.  What  is  uniformly-accelerated  velocity  1 

Bodies  which  receive  uniform  accelerations  of  velocity, 
that  is,  equal  accelerations  in  equal  times,  are  said  to  have 
motions  uniformly  accelerated.  * 

7.  How  will  a  body  fall  by  the  attraction  of  gravitation  ? 
If  a  body  fall  freely  towards  the  earth,  by  the  attraction 

of  gravitation,  it  will  descend  in  a  line  perpendicular  to  its 
surface. 

8.  How  far  will  it  fall  in  the  first  second^  and  how  far 
in  each  succeeding  second  ?  What  kind  of  velocity  will  it 
have  ? 

In  the  first  second  it  will  fall  through  16  feet;  in  the 
second  second,  having  the  velocity  already  acquired,  and 
being  still  acted  on  by  the  force  of  gravity,  it  will  descend 
through  48  feet ;  in  the  third  second  it  will  descend  through 
80  feet;  in  the  fourth  second  through  112  feet,  and  so  on, 
adding  to  its  velocity  in  every  additional  second.  This  is 
a  motion  uniformly  accelerated,  for  the  velocity  is  equally 
increased  in  each  second  of  time. 


LAWS    OF    MOTION,    ETC.  221 

9.  What  is  momentum  ? 

Momentum  is  the  force  with  which  a  body  in  motion 
would  strike  against  another  body.  If  a  body  of  a  given 
weight,  say  10  pounds,  were  moving  at  the  rate  of  30  feet 
per  second,  and  another  body  of  the  same  weight  were  to 
move  twice  as  fast,  the  last  would  have  double  the  mo- 
mentum of  the  first. 

10.  On  what  does  the  momentum  of  a  body  depend? 
When  the  bodies  are  of  a  given  weight,  the  momentum 

will  depend  on  the  velocity.  But  if  two  unequal  bodies 
move  with  the  same  velocity,  their  momentum  will  depend 
upon  their  weight.  Hence,  the  momentum  of  a  body  will 
depend  on  its  weight  and  velocity;  that  is^it  will  be  equal 
to  the  weight  multiplied  by  the  velocity. 

If  the  weight  of  a  body  be  represented  by  5,  and  its  ve- 
locity by  6,  its  momentum  will  be  5x6  =  30. 

If  the  weight  of  a  body  be   represented  by  8,  and  its 
velocity  by  2,  its  momentum  will  be  represented  by 
16x2  =  32. 

11.  What  are  action  and  reaction,  and  how  do  they  com- 
pare with  each  other? 

When  a  body  in  motion  strikes  against  another  body, 
it  meets  with  resistance.  The  force  of  the  moving  body 
is  called  action,  and  the  resistance  offered  by  the  body 
struck  is  called  reaction ;  and  it  is  a  general  principle,  that 
action  and  reaction  are  equal.  Thus,  if  you  strike  a  nail 
with  a  hammer,  the  action  of  the  hammer  against  the  nail 
is  just  equal  to  the  reaction  of  the  nail  against  the  ham- 
mer. Also,  if  a  body  fall  to  the  earth,  by  the  attraction 
of  gravitation,  the  action  of  the  body  when  it  strikes  the 
earth  is  just  equal  to  the  reaction  of  the  earth  against  the 
body. 


222 


BOOK    VII. SECTION    II. 


12.   What  is  the  centre  of  gravity? 

The  centre  of  gravity  is  that  point  of  a  body  about  which 
all  the  parts  will  exactly  balance  each  other.  Hence,  if 
the  centre  of  gravity  be  supported,  the  body  will  not  fall, 
for  all  the  parts  will  balance  each  other  about  the  centre 
of  gravity. 


13.  Is  the  centre  of  gravity  changed  by  changing  the  po- 
sition of  a  body  ? 

The  centre  of  gravity  of  a  body  is  not  changed  by 
changing  the  position  of  the  body.  Thus,  if  a  body  be 
suspended  by  a  cord,  attached  at  its  centre  of  gravity,  it 
will  remain  balanced,  in  every  position  of  the  body. 


14.  If  two  equal  bodies  are  joined  by  a  bar,  where  will  be 
the  centre  of  gravity  ? 

If  we  hav^e  two  equal  bodies 
A  and  B,  connected  together  by 
a  bar  AB,  the  centre  of  gravity 


will  be  at  C,  the  middle  point  of  AB,  and  about  this  point 
the  bodies  will  exactly  balance  each  other. 


15.  If  the  bodies  are  unequal,  where  will  it  be  found? 

If  we  have  two  unequal 
bodies  A  and  B,  the  cen- 
tre of  gravity  C  will  be 
nearer  the  larger  body  than 
the  smaller,  and  just  as 
much  nearer  as  the  larger 

body  exceeds  the  smaller.     Thus,  if  B  is  three  times  greater 
than  A,  then  BC  will  be  one-third  of  AC. 


CENTRE    OF    GRAVITY. 


223 


16.  If  one  body  is  very  large  in  comparison  with  the  other , 
what  will  take  place? 

If  one  of  the 
bodies  is  very  large 
in  comparison  with 
the  other,  the  cen- 
tre of  gravity  may 
fall  within  the  larger 

body.     Thus,  the  centre  of  gravity  of  the  bodies  A  and  B 
falls  at  C  • 

17.  What  is  the  line  of  direction  of  the  centre  of  gravity  ? 
The  vertical  line   drawn  through  the   centre  of  gravity, 

is  called  the  line  of  direction  of  the  centre  of  gravity. 

18.  If  this  line  is  supported,  will  the  hody  fall  1 
If  the  line  of  direction  of  the  centre 

of  gravity  falls  within  the  base  on 
which  the  body  stands,  the  body  will 
be  supported ;  but  if  the  line  falls  with- 
out the  base,  the  body  will  fall.  Thus, 
if  in  a  wine  glass,  the  centre  of  gravity 
be  at  C,  the  glass  will  fall  the  moment  the  line  CD  falls 
without  the  base. 

19.  If  the  line  of  direction  of  the   centre  of  gravity  falls 
near  the  base,  is  the  body  likely  to  fall  ?     Give  the  illustration. 

Let  us  suppose  a  cart  on  in- 
clined ground  to  be  loaded  with 
stone,  so  that  the  centre  of  gravity 
of  the  mass  shall  fall  at  C.  In 
this  position  the  line  of  direction 
CD  falls  within  the  base,  and  the 
cart  will  stand.  But  if  the  cart 
be  loaded  with  hay,  so  as  to  bring 


b  n 


224  BOOK    VII. SECTION    III. 

the  centre  of  gravity  at  A,  the  line  of  direction  AB  will 
fall  without  the  base,  and  the  cart  will  be  upset. 


SECTION   III. 


OF    THE    MECHANICAL    POWERS. 


1.  How  .many  mechanical  powers  are  there,  and  what  are 
they  ? 

There  are  six  simple  machines,  which  are  called  Mechan- 
ical powers.  They  are  the  Lever,  the  Pulley,  the  Wheel  and 
Axle,  the  Inclined  Plane,  the  Wedge,  and  the  Screw. 

2.  What  four  things  must  be  considered,  in  order  to  un- 
derstand the  power  of  a  machi?ie  ? 

1st.  The  power  or  force  which  acts.  This  consists  in 
the  effort  of  men  or  horses,  of  weights,  springs,  steam,  &c. 

2d.  The  resistance  which  is  to  be  overcome  by  the 
power.     This  generally  is  a  weight  to  be  moved. 

3d.  We  are  to  consider  the  centre  of  motion,  or  ful- 
crum, which  means  a  prop.  The  prop  or  fulcrum  is  the 
point  about  which  all  the  parts  of  the  machine  move. 

4th.  We  are  to  consider  the  respective  velocities  of  tht, 
power  and  resistance. 

3.  When  is  a  machine  said  to  he  in  equilibrium? 

A  machine  is  said  to  be  in  equilibrium  when  the  resist 
ance  exactly  balances  the  power,  in  which  case  all  the 
parts  of  the  machine  are  at  rest. 

4.  What  is  a  Lever? 

The  lever  is  a  straight  bar  of  wood  or  metal,  which 
moves  around  a  fixed  point  called  the  fulcrum. 


OF    THE    MECHANICAL    POWERS. 


225 


5.  How  many  kinds  of  levers  are  there  1 
There  are  three  kinds  of  levers : 
1st.   When  the  fulcrum  is 

between   the  weight  and  the 

power. 


2d.  When  the  weight  is 
between  the  power  and  the 
fulcrum. 


3d.  When  the  power  is 
between  the  fulcrum  and  the 
weight. 


6.  What  are  the  arms  of  a  lever  ? 

The  parts  of  the  lever,  from  the  fulcrum  to  the  weight 
and  power,  are  called  the  arms  of  the  lever. 

7.  When  is  an  equilibrium  produced  in  the  lever? 

An  equilibrium  is  produced  in  all  the  levers  when  the 
weight,  multiplied  by  its  distance  from  the  fulcrum,  is  equal 
to  the  product  of  the  power  multiplied  by  its  distance  from 
the  fulcrum.     That  is, 

The  weight  is  to  the  power,  as  the  distance  from  the  power 
to  the  fulcrum,  to  distance  from  the  weight  to  the  fulcrum. 


EXAMPLES. 


1.  In  a  lever  of  the  first  kind  the  fulcrum  is  placed  at 
the  middle  point:  what  power  will  be  necessary  to  bab 
ance  a   weight  of  40  pounds  ? 

10* 


226  LOOK    VII. SECTION    III. 

2.  In  a  lever  of  the  second  kind,  the  weight  is  placed 
at  the  middle  point :  what  power  will  be  necessary  to  sus- 
tain a  weight  of  50  lbs.  ? 

3.  In  a  lever  of  the  third  kind,  the  power  is  placed  at 
the  middle  point:  what  power  will  be  necessary  to  sus- 
tain a  weight  of  25  lbs.  ? 

4.  A  lever  of  the  first  kind  is  8  feet  long,  and  a  weight 
of  60  lbs.  is  at  a  distance  of  2  feet  from  the  fulcrum  • 
what  power  will  be  necessary  to  balance  it? 

Ans.  20  lbs. 

5.  In  a  lever  of  the  first  kind,  that  is  6  feet  long,  a 
weight  of  200  lbs.  is  placed  at  1  foot  from  the  fulcrum: 
what  power  will  balance  it? 

Ans.  40  lbs. 

6.  In  a  lever  of  the  first  kind,  like  the  common  steel- 
yard, the  distance  from  the  weight  to  the  fulcrum  is  one 
inch:  at  what  distance  from  the  fulcrum  must  the  poise 
of  1  lb.  be  placed,  to  balance  a  weight  of  1  lb.?  A  weight 
of  H  lbs.  ?     Of  2  lbs.  ?     Of  4  lbs.  ? 

7.  In  a  lever  of  the  third  kind,  the  distance  from  the 
fulcrum  to  the  power  is  5  feet,  and  from  the  fulcrum  to 
the  weight  8  feet :  what  power  is  necessary  to  sustain  a 
weight  of  40  lbs.  ? 

Ans.  64  lbs. 

8.  In  a  lever  of  the  third  kind,  the  distance  from  the 
fulcrum  to  the  weight  is  12  feet,  and  to  the  power  8  feet: 
what  power  will  be  necessary  to  sustain  a  weight  of  100 
lbs.  I 

Ans.  150  lbs. 

8.  How  are  the  equilibriums  of  levers  affected  by  consider- 
ing their  weight  ? 

Ill  levers  of  the  first  kind,  the  weight  of  the  ievcj  gener- 


OF    THE    PULLEY. 


227 


ally  adds  to  the  power,  but  in  the  second  and  third  kinds, 
the  weight  goes  to  diminish  the  effect  of  the  power. 

9.    What  has  been  stated  in  the  previous  examples  ?      What 
is  necessary  that  the  machine  may  move  ? 

In  the  previous  examples,  we  have  stated  the  circum 
stances  under  which  the  power  will  exactly  sustain  the 
weight.  In  order  that  the  power  may  overcome  the  re- 
sistance, it  must  of  course  be  somewhat  increased.  The 
lever  is  a  very  important  mechanical  power,  being  much 
used,  and  entering  indeed  into  all  the  other  machines.  . 


OF    THE    PIJLLEY. 


10.   What  is  a  pulley  ? 

The  pulley  is  a  wheel,  having  a  groove 
cut  in  its  circumference,  for  the  purpose 
of  receiving  a  cord  which  passes  over  it. 
When  motion  is  imparted  to  the  cord,  the 
pulley  turns  around  its  axis,  which  is 
generally  supported  by  being  attached  to 
a  beam  above. 


:t'  ;;"r> 


CI 


11.  How  many  kinds  of  pulleys  are  there? 
Pulleys   are   divided   into  two  kinds,  fixed   pulleys   and 
moveable  pulleys. 


12.  Does  a  fixed  pulley  increase  the  power? 

When  the  pulley  is  fixed,  it  does  not  increase  the  power 
which  is  applied  to  raise  the  weight,  but  merely  changos 
the  direction  in  which  it  acts. 


228 


BOOK    VII. SECTION    III. 


13.  Does  a  moveable  pulley  give  any  ad- 
vantage in  power? 

A  moveable  pulley  gives  a  mechanical 
advantage.  Thus,  in  the  moveable  pulley, 
the  hand  which  sustains  the  cask  does  not 
actually  support  but  one  half  the  weight 
of  it;  the  other  half  is  supported  by  the 
hook  to  which  the  other  end  of  the  cord 
is  attached. 


14.  Will  an  advantage  *he  gained  by  several 
moveable  pulleys  ?     What  will  be  lost  ? 

If  we  have  several  moveable  pulleys  the 
advantage  gained  is  still  greater,  and  a  very 
heavy  weight  may  be  raised  by  a  small  power. 
A  longer  time,  however,  will  be  required,  than 
with  a  single  pulley.  It  is  indeed  a  general 
principle  in  machines,  that  what  is  gained  in 
power  is  lost  in  time,  and  this  is  true  for  all 
machines. 


15.  Is  there  an  actual  loss  of  power?  What  does  it  arise 
from? 

There  is  also  an  actual  loss  of  power,  viz.,  the  resist- 
ance of  the  machine  to  motion,  arising  from  the  rubbing 
of  the  parts  against  each  other,  which  is  called  the  fric- 
tion of  the  machine.  This  varies  in  the  different  machines, 
but  must  always  be   allowed  for,  in  calculating  the  powei 


OF    THE    PULLEY.  229 

necessary  to  do  a  given  work.  It  would  be  wrong,  how- 
ever, to  suppose  that  the  loss  was  equivalent  to  the  gain, 
and  that  no  advantage  is  derived  from  the  mechanical 
powers.  We  are  unable  to  augment  our  strength,  but  by 
the  aid  of  science  we  so  divide  the  resistance,  that  by  a 
continued  exertion  of  power  we  accomplish  that  which  it 
would  be  impossible  to  effect  by  a  single  effort. 

If  in\attaining  this  result  we  sacrifice  time,  we  cannot 
but  see  that  it  is  most  advantageously  exchanged  for  power. 

16.  In  the  moveable  pulley,  what  proportion  exists  between 
the  power  and  the  weight? 

It  is  plain  that,  in  the  moveable  pulley,  all  the  parts  of 
the  cord  will  be  equally  stretched,  and  hence,  each  cord 
running  from  pulley  to  pulley  will  bear  an  equal  part  of 
the  weight ;  consequently,  the  power  will  always  be  equal  to 
the  weight,  divided  by  the  number  of  cords  which  reach  from 
pulley  to  pulley. 


EXAMPLES. 

1.  In  a  single  immoveable  pulley,  what  power  will  sup- 
port a  weight  of  60  lbs.  1 

2.  In  a  single  moveable  pulley,  what  power  will  support 
a  weight  of  80  lbs.  ? 

3.  In  two  moveable  pulleys  with  5  cords,  (see  last  fig.,) 
what  power  will  support  a  weight  of  100  lbs.? 

Ans.  20  lbs 


230  BOOK    VII. — SECTION    III. 


VIIEEL    AND    AXLE. 

17.  Of  what   is   the  wheel  and  axle  composed?     How 
the  axle  supported? 

This  machine   is   com-  / — n, 

posed  of  a  wheel  or  crank,  / 

firmly  attached  to  a  cylin- 
drical axle.  The  axle  is 
supported  at  its  ends  by 
two  pivots,  which  are  of 
less  diameter  than  the  axle 
around  which  the  rope  is 
coiled,  and  which  turn 
freely  about  the  points  of 
support. 

18.  What  is  the  proportion  between  the  power  and  weight? 
In  order  to  balance  the  weight,  we  must  have, 

The  power  to  the  weight,  as  the  radius  of  the  axle  to  the 
length  of  the  crank,  or  radius  of  the  wheel. 

EXAMPLES. 

1.  What  must  be  the  length  of  a  crank  or  radius  of  a 
wheel,  in  order  that  a  power  of  40  lbs.  may  balance  a 
weight  of  600  lbs.,   suspended  from  an  axle  of  6  inches 

radius?  Ans.lLft. 

2,  What  must  be  the  diameter  of  an  axle,  that  a  power 
of  100  lbs.  applied  at  the  circumference  of  a  wheel  of  6 
feet  diameter  may  balance  400  lbs.? 

Ans.  l\ft. 


INCLINED    PLANE.  231 


INCLINED    PLANE. 


19.  What  is  an  inclined  plane? 

The  inclined  plane  is  nothing  more  than  a  slope  or  de- 
clivity, which  is  used  for  the  purpose  of  raising  weights. 
It  is  not  difficult  to  see  that  a  weight  can  be  forced  up 
an  inclined  plane  more  easily  than  it  can  be  raised  in  a 
vertical  line.  But  in  this,  as  in  the  other  machines,  the 
advantage  is  obtained  by  a  partial  loss  of  power. 

20.  What  proportion  exists  between  the  power  and  the 
weight,  when  they  are  in  equilibrium? 

If  a  weight  W  be 
supported  on  the  in- 
clined plane  ABC  by 
a  cord  passing  over  a 
pulley  at  F,  and  the 
cord  from  the  pulley  to 

the  weight  be  parallel  to  the  length  of  the  plane  AB,  the 
power  P  will  balance  the  weight   W,  when 

P  :    W  :   :  height  BC  :  length  AB. 

It  is  evident  that  the  power  ought  to  be  less  than  the 
weight,  since  a  part  of  the  weight  is  supported  by  the 
plane. 

EXAMPLES. 

1.  The  length  of  a  plane  is  30  feet,  and  its  height  6 

feet:  what  power  will   be  necessary  to  balance  a  weight 

of  200  lbs.?  ,        ACi  „ 

Ans.  40  lbs. 

2.  The  height  of  a  plane  is  10  feet,  and  the  length  20 
feet  :  what  weight  will  a  power  of  50  lbs.  support  ? 

Ans.  100  lbs 


232 


BOOK    VII. SECTION    III. 


3.  The  height  of  a  plane  is  15  feet,  and  length  45  feet 
what  power  will  sustain  a  weight  of  180  lbs.  ? 

Ans.  60  lbs. 

THE    WEDGE. 

21.  What  is  the  wedge,  and  what  is  it  used  for 

The  wedge  is  composed  of  two  in- 
clined planes,  united  together  along 
their  bases,  and  forming  a  solid  A  CB. 
It  is  used  to  cleave  masses  of  wood 
or  stone.  The  resistance  which  it 
overcomes  is  the  attraction  of  cohe- 
sion of  the  body  which  it  is  employed 
to  separate.  The  wedge  acts  principally  by  being  struck 
with  a  hammer  or  mallet  on  its  head,  and  very  little  effect 
can  be  produced  with  it,  by  mere  pressure. 

All  cutting  instruments  are  constructed  on  the  principle 
of  the  inclined  plane  or  wedge.  Such  as  have  but  one 
sloping  edge,  like  the  chisel,  may  be  referred  to  the  in- 
clined plane ;  and  such  as  have  two,  like  the  axe  and  the 
knife,  to  that  of  the  wedge. 

THE    SCREW. 

22.  Of  how  many  parts ■  is  the  screw  composed  ?  Describe 
Us  parts  and  uses. 

The  screw  is  composed  of 
two  parts,  the  screw  S,  and 
the  nut  N. 

The  screw  S  is  a  cylinder 
with  a  spiral  projection  wind- 
ing around  it,  called  the  thread. 
The  nut  N  is  perforated  to  ad- 
mit the  screw,  and  within  it  is 
a  groove  into  which  the  thread 
of  the  screw  fits  closely. 


THE    SCREW.  233 

The  handle  Z>,  which  projects  from  the  nut,  is  a  lever 
which  works  the  nut  upon  the  screw.  The  power  of  the 
screw  depends  on  the  distance  between  the  threads.  The 
closer  the  threads  of  the  screw,  the  greater  will  be  the 
power,  but  then  the  number  of  revolutions  made  by  the 
handle  D  will  also  be  proportionally  increased ;  so  that  we 
return  to  the  general  principle — what  is  gained  in  power 
is  lost  in  time.  The  power  of  the  screw  may  also  be  in- 
creased by  lengthening  the  lever  attached  to  the  nut. 

The  screw  is  used  for  compression,  and  to  raise  heavy 
weights.  It  is  used  in  cider  and  wine  presses,  in  coining, 
and  for  a  variety  of  other  purposes. 

GENERAL    REMARKS. 

All  machines  are  composed  of  one  or  more  of  the  six 
machines  which  we  have  described.  We  should  remem- 
ber, that  friction  diminishes  very  considerably  the  power 
of  machines. 

There  are  no  surfaces  in  nature  which  are  perfectly 
smooth.  Polished  metals,  although  they  appear  smooth,  are 
yet  far  from  being  so.  If,  therefore,  the  surfaces  of  two 
bodies  come  into  contact,  the  projections  of  the  one  will 
fall  into  the  hollow  parts  of  the  other,  and  occasion  more 
or  less  resistance  to  motion.  In  proportion  as  the  surfaces 
of  bodies  are  polished,  the  friction  is  diminished,  but  it  is 
always  very  considerable,  and  it  is  computed  that  it  gen- 
erally destroys  one-third •  the  power  of  the  machine. 

Oil  or  grease  is  generally  used  to  lessen  the  friction. 
It  fills  up  the  cavities  of  the  rubbing  surfaces,  and  thus 
makes  them  slide  more  easily  over  each  other. 


234  BOOK    VII. SECTION    IV. 


SECTION  IV. 

OF    SPECIFIC    GRAVITY. 

1.  What  is  the  specific  gravity  of  a  body? 

The  specific  gravity  of  a  body  is  the  relation  which  the 
weight  of  a  given  magnitude  of  that  body  bears  to  the 
weight  of  an  equal  magnitude  of  a  body  of  another  kind 

2.  When  is  one  body  said  to  be  specifically  heavier  than 
another  ? 

If  two  bodies  are  of  the  same  bulk,  the  one  which  weighs 
the  most  is  said  to  be  specifically  heavier  than  the  other. 
On  the  contrary,  one  body  is  said  to  be  specifically  lighter 
than  another,  when  a  certain  bulk  or  volume  of  it  weighs 
less  than  an  equal  bulk  of  that  other. 

Thus,  if  we  have  two  equal  spheres,  each  one  foot  in 
diameter,  the  one  of  lead  and  the  other  of  wood,  the  leaden 
one  will  be  found  to  be  heavier  than  the  wooden  one  ;  and 
hence,  its  specific  gravity  is  greater.  On  the  contrary,  the 
wooden  sphere  being  lighter  than  the  leaden  one,  its  specific 
gravity  is  less. 

3.  What  does  the  greater  specific  gravity  indicate  ?  What 
is  density  ? 

The  greater  specific  gravity  of  a  body  indicates  a  greater 
quantity  of  matter  in  a  given  bulk,  and  consequently  the 
matter  must  be  more  compact,  or  the  particles  nearer  to- 
gether. This  closeness  of  the  particles  is  called  density. 
Hence,  if  two  bodies  are  of  equal  bulk  or  volume,  their 
weights  or  specific  gravities  will  be  proportional  to  their 
densities.  *■ 


OF    SPECIFIC    GRAVITY.  235 

4.  If  two  bodies  are  of  the  same  specific  gravity,  how  will 
the  weights  be? 

If  two  bodies  are  of  the  same  specific  gravity,  or  density, 
their  weights  will  be  proportional  to  their  bulks. 

5.  If  a  body  be  immersed  in  a  fluid,  what  will  take  place  f 
If  the  body  is  specifically  heavier  than  the  fluid,  it  will 

sink  on  being  immersed.  It  will,  however,  descend  less 
rapidly  through  the  fluid  than  through  the  air,  and  less 
power  will  be  required  to  sustain  the  body  in  the  fluid  than 
out  of  it.  Indeed,  it  will  lose  as  much  of  its  weight  as  is 
equal  to  the  weight  of  a  quantity  of  fluid  of  the  same  bulk. 
If  the  body  is  of  the  same  specific  gravity  with  the  fluid, . 
it  loses  all  its  weight,  and  requires  no  force  but  the  fluid 
to  sustain  it.  If  it  be  lighter,  it  will  be  but  partially  im- 
mersed, and  a  part  of  the  body  will  remain  above  the  sur- 
face of  the  fluid. 

6.  What  do  we  conclude  from,  the  preceding  article  ? 
1st.  That  when  a  heavy  body  is  weighed  in  a  fluid,  its 

weight  will  express  the  difference  between  its  true  weight 
and  that  of  an  equal  bulk  of  the  fluid. 

2d.  If  the  body  have  the  same  specific  gravity  with  the 
fluid,  its  weight  will  be  nothing. 

3d  If  the  body  be  lighter  than  the  fluid,  it  will  require 
a  force  equal  to  the  difference  between  its  own  weight 
and  that  of  an  equal  bulk  of  the  fluid  to  keep  it  entirely 
immersed,  that  is,  to  overcome  its  tendency  to  rise. 

7.  What  is  necessary  in  comparing  the  weights  of  bodies  ? 
In  comparing  the  weights  of  bodies,  it  is  necessary  to 

take  some  one  as  a  standard,  with  which  to  compare  all 
otheie. 


236  BOOK    VII. SECTION    IV. 

8.  What  is  generally  taken  as  the  standard? 
Rain-water  is  generally  taken  as  this  standard. 

9.  What  is  the  weight  of  a  cubic  foot  of  rain-water  ? 

A  cubic  foot  of  rain-water  is  found,  by  repeated  experi- 
ments, to  weigh  621  pounds,  avoirdupois,  or  1000  ounces. 
Now,  since  a  cubic  foot  contains  1728  cubic  inches,  it  fol- 
lows that  one  cubic  inch  weighs  .03616898148  of  a  pound. 
Therefore,  if  the  specific  gravity  of  any  body  be  multiplied 
by  .03616898148,  the  product  will  be  the  weight  of  a  cubic 
inch  of  that  body  in  pounds  avoirdupois.  And  if  this  weight 
be  then  multiplied  by  175,  and  the  product  divided  by  144, 
the  quotient  will  be  the  weight  of  a  cubic  inch  in  pounds 
troy;  since  144  lbs.  avoirdupois  is  just  equal  to  175  lbs. 
troy. 

10.  How  will  the  specific  gravity  of  a  body  be  to  that  of 
the  fluid  in  which  it  is  immersed  1 

Since  the  specific  gravities  of  bodies  are  as  the  weights 
of  equal  bulks,  the  specific  gravity  of  a  body  will  be  to 
the  specific  gravity  of  a  fluid  in  which  it  is  immersed,  as 
the  true  weight  of  the  body  to  the  weight  lost  in  weighing 
it  in  the  fluid.  Hence,  the  specific  gravities  of  different  fluids 
are  to  each  other  as  the  weights  lost  by  the  same  solid  im~ 
mersed  in  them. 

11.  How  do  you  find  the  specific  gravity  of  a  body,  when 
tlie  body  is  heavier  than  water? 

1st.  Weigh  the  body  first  in  air  and  then  in  rain-water, 
and  take  the  difference  of  the  weights,  which  is  the  weight 
lost. 

*  2d.  Then  say,  as  the  weight  lost  is  to  the  true  weight, 
£0  is  the  specific  gravity  of  the  water  to  the  specific  grav« 
ity  of  the  body. 


OF    SPECIFIC    GRAVITY.  237 


EXAMPLES. 


1.  A  piece  of  platina  weighs  70.5588  lbs.  in  the  air, 
and  in  water  only  66.9404  lbs. :  what  is  its  specific  gravity, 
that  of  water  being  taken  at  1000? 

First,  70.5588  —  66.9404  =  3.6184  lost  in  water. 

Then,    3.6184  :  70.5588  :  :   1000  :   19500,   which   is    the 

specific  gravity,  or  weight  of  a  cubic  foot  of  platina. 

2.  A  piece  of  stone  weighs  10  lbs.  in  air,  but  in  water 
only  6f  lbs. :   what  is  its  specific  gravity  ? 

Ans.  3077. 

12.  How  do  you  find  the  -specific  gravity  of  a  body  when 
it  is  lighter  than  water? 

1st.  Attach  another  body  to  it  of  such  specific  gravity, 
that  both  may  sink  in  the  water  as  a  compound  mass. 

2d.  Weigh  the  heavier  body  and  the  compound  mass 
separately,  both  in  water  and  in  open  air,  and  find  how 
much  each  loses  by  being  weighed  in  water. 

3d.  Then  say,  as  the  difference  of  these  losses  is  to  the 
weight  of  the  lighter  body  in  the  air,  so  is  the  specific 
gravity  of  water  to  the  specific  gravity  of  the  lighter  body. 

EXAMPLES. 

1.  A  piece  of  elm  weighs  15  lbs.  in  open  air.  A  piece 
of  copper  which  weighs  18  lbs.  in  air  and  16  in  water  is 
attached  to  it,  and  the  compound  weighs  6  lbs.  in  water- 
what  is  the  specific  gravity  of  the  elm? 

Copper.  Compound. 

18  in  air.  33  in  air. 

16  in  water.  6  in  water 

2  loss.  27  loss. 

Then,  27  —  2  =  25  =  difference  of  losses. 


238  BOOK    VII. — SECTION    IV. 

Then,  as  25  :  15  :  :  1000  :  600,  which  is  the  specific 
gravity  of  the  elm. 

2.  A  piece  of  cork  weighs  20  lbs.  in  air,  and  a  piece  of 

granite  weighs  120  lbs.  in  air,  and  80  lbs.  in  water.     When 

the    granite   is   attached  to  the   cork   the   compound  mass 

weighs  16f  lbs.  in  water:   what  is  the  specific  gravity  of 

the  cork?  A        0.n 

Ans.  240. 

13.  How  do  you  find  the  specific  gravity  of  fluids? 

1st.  Weigh  any  body  whose  specific  gravity  is  known, 
both  in  the  open  air,  and  in  the  fluid,  and  take  the  differ- 
ence, which  is  the  loss  of  weight. 

2d.  Then  say,  as  the  true  weight  is  to  the  loss  of  weight, 
so  is  the  specific  gravity  of  the  solid  to  the  specific  grav- 
ity of  the  fluid. 

EXAMPLES. 

1.  A  piece  of  iron  weighs  298.1  ounces  in  the  air,  and 
259.1  ounces  in  a  fluid;  the  specific  gravity  of  the  iron  is 
7645 :   what  is  the  specific  gravity  of  the  fluid  1 

First,  298.1  —  259.1  =  39  loss  of  weight: 
Then,  298.1  :  39  ::  7645  :  1000,  which  is  the  specific  grav- 
ity of  the  fluid :  hence  the  fluid  is  water. 

2.  A  piece  of  lignumvitas  weighs  42f  ounces  in  a  fluid, 
and  166f  ounces  out  of  it:  what  is  the  specific  gravity  of 
the  fluid — that  of  lignumvitse  being  1333  ? 

Ans.  991,  which  shows  the  fluid  to  be  liquid  turpentine 
or  Burgundy  wine. 

Note. — In  a  similar  manner  the  specific  gravities  of  all 
liquids  may  be  found  from  the  following  table 


OF    SPECIFIC    GRAVITIES. 


239 


TABLE    OF    SPECIFIC    GRAVITIES. 


Sp.gr.  wt.  cub.  in. 

Platina,  hammered  20.331 

11.777 

Platina 

19.500  - 

11.285 

Pure  cast  gold 

•  19.258  - 

11.145, 

Mercury 

13.568  - 

7.872 

Cast  lead 

11.352  - 

6.569 

Pure  cast  silver 

10.474  - 

6.061 

Cast  copper 

8.788  - 

5.085 

Cast  brass 

8.395  - 

4.856 

Hard  steel 

7.816  - 

4.523 

Cast  cobalt 

7.811  - 

4.520 

Cast  nickel 

7.807  - 

4.513 

Bar  iron 

7.788  - 

4.507 

Cast  tin 

7.291  - 

4.219 

Cast  iron 

7.207  - 

4.165 

Cast  zinc 

7.190  - 

4.161 

w 

t.  cub.  ft. 
lbs. 

Limestone 

3.179  - 

198.68 

White  glass 

2.892 

Chalk 

2.784  - 

174.00 

Marble 

2.742  - 

171.38 

Alabaster 

2.730 

Pearl 

2.684 

Slate 

2.672  - 

167.00 

Pebble 

2.664  - 

166.50 

Green  glass 

2.642 

Flint  and  spar 

2.594  - 

162.12 

Common  stone 

2.520  - 

157.50 

Paving  stones 

2.416  - 

lvl.00 

Sulphur 

2.033  - 

127.06 

Brick 

2.000  - 

125.00 

Ivory 

1.822 

Bone  of  an  ox 

1.659 

Honey 

1456 

Lignumvitm 

1.333  - 

83.31 

Ebony  1 

Oak,  60  years  old        1 

Amber  1 

Beer  1 

Milk  1 

Sea  water  1 

Distilled  water  1 

Liquid  turpentine 

Burgundy  wine 

Camphor 

Oak,  English 

Bees'  wax 

Tallow 

Olive  oil 

Logwood 

Box,  French 

Wax 

Oak,  Canadian  • 

Alder 

Apple  tree 

Ash  and  Dantzic  oak 

Maple  and  Riga  fir 

Cherry  tree 

Beech 

Elder  tree 

Walnut 

Pear  tree 

Pitch  pine 

Cedar  • 

Mahogany 

Elm  and  West  India  fir 

Larch 

Poplar 

Cork 

Air  at  the  earth's  surf. 


gi   wt. 
331  - 
170  - 
.078 
034 
030 
028 
.000 
.991 
.991 
.989 
.970  - 
.965 
.945 
.915 
.913  - 
.912  - 
.897 
.872  - 
.800  - 
.793  - 
.760  - 
.750  - 
.715  - 
.696  - 
.695  - 
.671  - 
.661  - 
.660  - 
.596  - 
.560  - 
.556  - 
.544  - 
.383  - 
.240  - 
.00lf 


cub.  ft. 
83.18 
73.12 


60.62 


57.06 
57.00 

54.50 
50.00 
49.56 
47.50 
46.87 
44.68 
43.50 
43.44 
41.94 
41.31 
41.25 
37.25 
35.00 
34.75 
34.00 
23.94 
15.00 


Rkmark. — In  the  table  of  specific  gravities,  the  cubic 
foot  of  water,  which  weighs  1000  ounces,  is  taken  as  the 
standard,  and  the  figures  in  the  column  of  specific  gravity 
show  how  many  times  each  substance  is  heavier  or  lighter 
than  water.  If  the  number  opposite  each  substance  be 
multiplied  by  1000,  the  product  will  be  the  weight  of  a 
cubic  foot  of  that  substance,  in  ounces.  The  other  column 
shows  the  weight  in  ounces  of  a  cubic  inch,  or  the  weight 
in  pounds  of  a  cubic  foot. 


240  BOOK    VII. — SECTION    IV. 

14.  How  do  you  find  the  solidity  of  a  body  when  its  weight 
and  specific  gravity  are  given  ? 

As  the  tabular  specific  gravity  of  the  body  is  to  its 
weight  in  ounces  avoirdupois,  so  is  1  cubic  foot  to  the  con- 
tent in  cubic  feet. 

EXAMPLES. 

1.  What  is  the  solid  content  of  a  block  of  marble,  that 
weighs  10  tons,  its  specific  gravity  being  2742? 

First,  10  tons  =  358400  ounces. 
Then.  2742  :  358400  ::  1  :  130,^,  which  is  the  content 
in  cubic  feet. 

Note. — If  the  answer  is  to  be  found  in  cubic  inches, 
multiply  the  ounces  by  1728. 

2.  How  many  cubic  inches  in  an  irregular  block  of 
marble,  which  weighs  112  pounds,  allowing  its  specific 
gravity  to  be  2520? 

3.  How  many  cubic  inches  of  gunpowder  are  there  in 
1  pound  weight,  its  specific  gravity  being  1745  ? 

Ans.  15f,  nearly. 

4.  How  many  cubic  feet  are  there  in  a  ton  weight  of 
dry  oak,  its  specific  gravity  being  925  ? 

Ans.  38m. 


OP    LOGARITHMS.  241 


BOOK   VIII. 


APPLICATIONS    OF   MATHEMATICS, 


SECTION   I. 


OF    LOGARITHMS. 


1.  The  logarithm  of  a  number  is  the  exponent  of  the  power 
to  which  it  is  necessary  to  raise  a  fixed  number,  in  order  to 
produce  the  first  number. 

This  fixed  number  is  called  the  base  of  the  system,  and  may- 
be any  number  except  1 :  in  the  common  system  10  is  assumed 
as  the  base. 

2.  If  we  form  those  powers  of  10,  which  are  denoted  by  entire 
exponents,  we  shall  have 

10°  =  1  101  =  10       ,       103  =  1000 

102  =  100     ,       104  =  10000,  <fcc.  &c. 
From  the  above  table,  it  is  plain,  that  0,  1,  2,  3,  4,  &c,  are  re- 
spectively the  logarithms  of  1,  10,  100,  1000,  10000,  &c. ;  we 
also  see  that  the  logarithm  of  any  number  between  1  and  10  is 
greater  than  0  and  less  than  1 :  thus 
Log  2=:  0.301030 
The  logarithm  of  any  number  greater  than  10,  and  less  than 
100,  is  greater  than  1  and  less  than  2  :  thus 

Log  50  =  1.698970 
11 


242  BOOK    VIII. SECTION    1. 

The  logarithm  of  any  number  greater  than  100,  and  less  than 
1000,  is  greater  than  2  and  less  than  3  :  thus 
Log  126  =  2.100371,  &c. 

If  the  above  principles  be  extended  to  other  numbers,  it  will 
appear,  that  the  logarithm  of  any  number,  not  an  exact  power 
of  ten,  is  made  up  of  two  parts,  an  entire  and  a  decimal  part. 
The  entire  part  is  called  the  characteristic  of  the  logarithm, 
and  is  always  one  less  than  the  number  of  places  of  figures  in  the 
given  number. 

3.  The  principal  use  of  logarithms,  is  to  abridge  numerical 
computations. 

Let  M  denote  any  number,  and  let  its  logarithm  be  denoted 
by  m  ;  also  let  N  denote  a  second  number  whose  logarithm  is 
n  ;  then  from  the  definition  we  shall  have 

iom=  m  (i)         ion  =  jv  (2) 

Multiplying  equations  (1)  and  (2),  member  by  member,  we 
have 

10m+"  =  MxN    or,  m-\-n  =  log  MxN ':     hence, 

The  sum  of  the  logarithms  of  any  two  numbers  is  equal  to 
the  logarithm  of  their  product. 

Dividing  equation  (1)  by  equation   (2),  member  by  member, 

we  have 

,    14       jf'  .        M      . 

10       =  «a    or,    m — n  =  log  -^:    hence, 

■  JSf  &  JSf 

The  logarithm  of  the  quotient  of  two  numbers,  is  equal  to  the 
logarithm  of  the  dividend  diminished  by  the  logarithm  of  the 
divisor. 

4.  Since  the  logarithm  of  10  is  1,  the  logarithm  of  the  product 
of  any  number  by  10,  will  be  greater  by  1  than  the  logarithm  of 


OF    LOGARITHMS.  243 

that  number;  also,  the  logarithm  of  any  number  divided  by 10, 
will  be  less  by  1  than  the  logarithm  of  that  number. 

Similarly,  it  may  be  shown  that  the  logarithm  of  any  number 
multiplied  by  a  hundred,  is  greater  by  2  than  the  logarithm  of 
that  number,  and  the  logarithm  of  any  number  divided  by  100 
is  less  by  2,  than  the  logarithm  of  that  number,  and  so  on 

EXAMPLES. 


log  327 

is 

2.514548 

log  32.7 

U 

1.514548 

log  3.27 

u 

0.514548 

log  .327 

u 

1.514548 

log  .0327 

u 

2.514548 

From  the  above  examples,  we  see,  that  in  a  number  composed 
of  an  entire  and  decimal  part,  we  may  change  the  place  of  the 
decimal  point  without  changing  the  decimal  part  of  the  logarithm; 
but  the  characteristic  is  diminished  by  1  for  every  place  that  the 
decimal  point  is  removed  to  the  left. 

In  the  logarithm  of  a  decimal,  the  characteristic  becomes  nega- 
tive, and  is  numerically  1  greater  than  the  number  of  ciphers  im- 
mediately after  the  decimal  point.  The  negative  sign  extends 
only  to  the  characteristic,  and  is  written  over  it  as  in  the  exam- 
ples given  above. 

TABLE    OF    LOGARITHMS. 

5.  A  table  of  logarithms,  is  a  table  in  which  are  written  the 
logarithms  of  all  numbers  between  1  and  some  given  number. 
The  logarithms  of  all  numbers  between  1  and  10,000  are  given 
in  the  annexed  table.  Since  rules  have  been  given  for  determin- 
ing the  characteristics  of  logarithms  by  simple  inspection,  it  has 
not  been  deemed  necessary  to  write  them  in  the  table,  the  deci- 


244  BOOK   VIII. — SECTION    I. 

mal  part  only  being  given.  The  characteristic,  however,  is  given 
for  all  numbers  less  than  100.     % 

The  left  hand  column  of  each  page  of  the  table,  is  the  column 
of  numbers,  and  is  designated  by  the  letter  N ;  the  logarithms 
of  these  numbers  are  placed  opposite  them  on  the  same  hori- 
zontal line.  The  last  column  on  each  page,  headed  D,  shows  the 
difference  between  the  logarithms  of  two  consecutive  numbers. 
This  difference  is  found  by  subtracting  the  logarithm  under  the 
column  headed  4,  from  the  one  in  the  column  headed  5  in  the 
same  horizontal  line,  and  is  nearly  a  mean  of  the  differences 
of  any  two  consecutive  logarithms  on  the  line. 

6.  To  find  from  the  table  the  logarithm  of  any  number. 

If  the  number  is  less  than  100,  look  on  the  first  page  of  the 
table,  in  the  column  of  numbers  under  N,  until  the  number  is 
found  :  the  number  opposite  is  the  logarithm  sought :  Thus 
log  9  =  0.954243 

7.  When  the  number  is  greater  than  100  and  less  than  10000 
Find  in  the  column  of  numbers,  the  first  three  figures  of  the 

given  number.  Then  pass  across  the  page  along  a  horizontal 
line  until  you  come  into  the  column  under  the  fourth  figure  of 
the  given  number :  at  this  place,  there  are  four  figures  of  the 
required  logarithm,  to  which  two  figures  taken  from  the  column 
marked  0,  are  to  be  prefixed. 

If  the  four  figures  already  found  stand  opposite  a  row  of  six 
figures  in  the  column  marked  0,  the  two  left  hand  figures  of 
the  six,  are  the  two  to  be  prefixed ;  but  if  they  stand  opposite 
a  row  of  only  four  figures,  you  ascend  the  column  till  you  find 
a  row  of  six  figures ;  the  two  left  hand  figures  of  this  row  are 
the  two  to  be  prefixed.     If  you  prefix  to  the  decima    part  thus 


OF    LOGARITHMS.  245 

found,  the  characteristic,  you   will  have   the  logarithm  sought: 
Thus, 

log     8979  =  3.953228 

log  .08979  =  2.953228 
If  however  in  passing  back  from  the  four  figures  found,  to  the 
0  column,  any  dots  be  met  with,  the  two  figures  to  be  prefixed 
must  be  taken  from  the  horizontal  line  directly  below  :    Thus, 

log   3098  =  3.491081 

log  30.98  =  1.491081 
If  the  logarithm  falls  at  a  place  where  the  dots  occur,  0  must 
be  written  for  each   dot,  and  the  two  figures  to  be  prefixed  are 
as  before  taken  from  the  line  below  :    Thus, 

log  2188  =  3.340047 

log  .2188  =  1.340047 

8.   When  the  number  exceeds  10,000. 

The  characteristic  is  determined  by  the  rules  already  given. 
To  find  the  decimal  part  of  the  logarithm.  Place  a  decimal 
point  after  the  fourth  figure  from  the  left  hand,  converting  the 
given  number  into  a  whole  number  and  decimal.  Find  the  loga- 
rithm of  the  entire  part  by  the  rule  just  given,  then  take  from 
the  right  hand  column  of  the  page,  under  D,  the  number  on  the 
same  horizontal  line  with  the  logarithm,  and  multiply  it  by  the 
decimal  part ;  add  the  product  thus  obtained  to  the  logarithm  al- 
ready found,  and  the  sum  will  be  the  logarithm  sought. 

If,  in  multiplying  the  number  taken  from  the  column  D,  the 
decimal  part  of  the  product  exceeds  .5  let  1  be  added  to  the  en- 
tire  part;  if  it  is  less  than  .5  the  decimal  part  of  the  product  is 
neglected. 


240  BOOK   VIII. SECTION    I. 

EXAMPLE. 

To  find  log  672887. 

The  characteristic  is  5. ;  placing  a  decimal  point  after  the 
fourth  figure  from  the  left,  we  have  6728.87.  The  decimal  part 
of  the  log  6728  is  .827886  and  the  corresponding  number  in  the 
column  D  is  65  ;  then  65X-87  as  56.55,  and  since  the  decimal 
part  exceeds  .5,  we  have  57  to  be  added  to  827886,  which  gives 
,827943 

or     log  672887     =  5.827943 
Similarly  log  .0672887  =  2.827943 

The  last  rule  has  been  deduced  under  the  supposition  that  the 
difference  of  the  numbers  is  proportional  to  the  difference  of 
their  logarithms,  which  is  sufficiently  exact  within  the  narrow 
limits  considered. 

In  the  above  example,  65  is  the  difference  between  the  loga- 
rithm of  672900  and  the  logarithm  of  672800,  that  is,  it  is  the 
difference  between  the  logarithms  of  two  numbers  which  differ  by 
100. 

We  have  then  the  proportion  100  :  87  :  :  65  :  56.55,  the 
number  to  be  added  to  the  logarithm  already  found. 

9.  To  find  from  the  table  the  number  corresponding  to  a 
given  logarithm. 

Search  in  the  columns  of  logarithms  for  the  decimal  part  of 
the  given  logarithm :  if  it  cannot  be  found  in  the  table,  take 
out  the  number  corresponding  to  the  next  less  logarithm  and 
set  it  aside.  Subtract  this  less  logarithm  from  the  given  loga- 
rithm, and  annex  to  the  remainder  as  many  zeros  as  may  be 
necessary,  and  divide  this  result  by  the  corresponding  number 
taken  from  the  column  marked  D,  continuing  the  division  as 
long  as  desirable  :  annex  the  quotient  to  the   number  set  aside. 


CF    LOGARITHMS.  247 

Point  off,  from  the  left  hand,  as  many  integer  figures  as  there  are 
units  in  the  characteristic  of  the  given  logarithm  increased  by 
1 ;  the  result  is   the  required  number. 

If  the  characteristic  is  negative,  the  number  will  be  entirely 
decimal,  and  the  number  of  zeros  to  be  placed  immediately  after 
the  decimal  point  will  be  equal  to  the  number  of  units  in  the 
characteristic  diminished  by  1. 

This  rule,  like  its  converse,  is  founded  on  the  supposition  that 
the  difference  of  the  logarithms  is  proportional  to  the  difference 
of  their  numbers  within  narrow  limits. 

EXAMPLE. 

Find  the  number  corresponding  to  the  logarithm  3.233568. 

The  decimal  part  of  the  given  logarithm  is    .233568 

The  next  less  logarithm  of  the  table  is  .233504    and    its 

corresponding  number  1712.  

Their  difference  is  64 


Tabular  difference  253)6400000(25 

Hence  the  number  sought         1712.25 

The  number  corresponding  to  3.233568  is  .00171225 

MULTIPLICATION    BY    LOGARITHMS. 

10,  When  it  is  required  to  multiply  numbers  by  means  of 
their  logarithms,  we  first  find  from  the  table  the  logarithms  of 
the  numbers  to  be  multiplied  ;  we  next  add  these  logarithms 
together,  and  their  sum  is  the  logarithm  of  the  product  of  the 
numbers  (Art.  3). 

The  term  sum  is  to  be  understood  in  its  algebraic  sense ; 
therefore,  if  any  of  the  logarithms  have  negative  characteristics, 
the  difference  between  their  sum  and  that  of  the  positive 
characteristics,  is  to  be  taken  ;  the  sign  of  the  remainder  is 
that  of  the  greater  sum. 


248  BOOK   VIII. — SECTION    L 

EXAMPLES. 

1.  Multiply  23.14  by  5.062. 

log  23.14  =  1.364363 

log  5.062  =  0.704322 

Product  117.1347  ....  2.068685 


2.  Multiply  3.902,  597.16  and  0.0314728  together. 

log    3.902  =  0.591287 

log  597.16  =  2.776091 

log  0.0314728  =  2.497936 

Product  73.3354  ....  1.865314 


Here  the  2  cancels  the  +  2,  and  the  1  carried  from  the  dec> 
mal  part  is  set  down. 

3.  Multiply  3.586,  2.1046,  0.8372,  and  0.0294,  together 
log     3.586  =  0.554610 
log  2.1046  ars  0.323170 
log  0.8372  =  1.922829 
log  0.0294  =  2.468347 

Product    0.1857615     .     .     1.268956 


In  this  example  the  2,  carried  from  the  decimal  part,  cancels 
2,  and  there  remains  1  to  be  set  down. 

DIVISION    OF    NUMBERS    BY    LOGARITHMS. 

1 1  When  it  is  required  to  divide  numbers  by  means  of  their 
logarithms,  we  have  only  to  recollect,  that  the  subtraction  of 
logarithms  corresponds  to  the  division  of  their  numbers  (Art.  3). 
Hence,  if  we  find  the  logarithm  of  the  dividend,  and  from  it  sub- 
tract the  logarithm  of  the  divisor,  the  remainder  will  be  the  loga- 
rithm of  the  quotient. 


OF    LOGARITHMS.  249 

This  additional  caution  may  be  added.  The  difference  of  the 
logarithms,  as  here  used,  means  the  algebraic  difference;  so 
that,  if  the  logarithm  of  the  divisor  have  a  negative  characteristic 
its  sign  must  be  changed  to  positive,  after  diminishing  it  by  the 
unit,  if  any,  carried  in  the  subtraction  from  the  decimal  part  of 
the  logarithm.  Or,  if  the  characteristic  of  the  logarithm  of  the 
dividend  is  negative,  it  must  be  treated  as  a  negative  number. 

EXAMPLES. 

1.  To  divide  24163  by  4567. 

log  24168  =  4.383151 

log     4567  =  3.659631 

Quotient      5.29078     ....     0.723520 

2.  To  divide  0.06314  by  .007241 

log      0.06314  ==  2.800305 

log    0.007241  =  3.859799 

Quotient     .     .     8.7198    .     .     .     .     0.940506 

Here,  1  carried  from  the  decimal  part  to  the  3  changes  it  to 
2,  which  being  taken  from  2,  leaves  0  for  the  characteristic. 

3.  To  divide  37.149  by  523.76 

log  37.149  =  1.5699*7 
log  523.76  =  2.719133 

Quotient     .     .     0.0709274      .      2.850814 

4.  To  divide  0.7438  by  12.9476 

log    0.7438  =  1.871456 
log  12.9476  =  1.112189 

Quotient      .      .      0.057447      .      .    "2.759267 

Here,  the  1  taken  from  1,  gives  2  for  a  result,  as  set  down 
1 1* 


250 


BOOK    VIII. SECTION    I. 


ARITHMETICAL    COMPLEMENT. 


12.  The  Arithmetical  complement  of  a  logarithm  is  the  num- 
ber which  remains  after  subtracting  the  logarithm  from  10. 
Thus,         .         .         1-9.274687  =  0.725313 
Hence,  0.725313    is  the  arithmetical  complement 

of  9.274687. 

13.  We  will  now  show  that,  the  difference  between  two  loga- 
rithms is  truly  found,  by  adding  to  the  first  logarithm  the 
arithmetical  complement  of  the  logarithm  to  be  subtracted,  and 
then  diminishing  the  sum  by  10. 

Let     a  =  the  first  logarithm 

b  =  the  logarithm  to  be  subtracted 
and  c  =  10  —  b  =  the  arithmetical  complement  of  6. 

Now  the  difference  between  the  two  logarithms  will  be  ex- 
pressed by  a— 6. 

But,  from  the  equation  c  —  10  —  6,  we  have 
c-10  =  —b 
hence,  if  we   place  for— b  its  value,  we  shall  have 

a  —  b  =  a-\-c — 10 
which  agrees  with   the  enunciation. 

When  we  wish  the  arithmetical  complement  of  a  logarithm, 
we  may  write  it  directly  from  the  table,  by  subtracting  the  left 
hand  figure  from  9,  then  proceeding  to  the  right,  subtract  each 
figure  from  9  till  we  reach  the  last  significant  figure,  which 
must  be  taken  from  10:  this  w:ll  be  the  same  as  taking  the 
logarithm  from  10. 


OP    LOGARITHMS.  251 

EXAMPLES. 

1.  From  3.2V4107     take     2.104729. 

By  common  method.  By  arith.  comp. 

3.274107  3.274107 

2.104729         its  ar.  comp.  7.895271 

Diff.      1.169378  Sum   1.169378  after  sub- 


tracting 10. 

Hence,  to  perform  division  by  means  of  the  arithmetical  com- 
plement we  have  the  following 

RULE. 

To  the  logarithm  of  the  dividend  add  the  arithmetical  com* 
plement  of  the  logarithm  of  the  divisor :  the  sum  after  subtract* 
ing  10,  will  be  the  logarithm  of  the  quotient. 

EXAMPLES. 

1.  Divide  327.5  by  22.07. 

log  327.5         .         .         .         2.515211 

log  22.07         ar.  comp.  8.656198 

Quotient     .     .'     14.839      .      .      .     ,   '  1.171409 

2.  Divide  0.7438  by  12.9476. 

log    0.7438 T.871456 

log  12.9476         ar.  comp.         8.887811 

Quotient     .     .     0.057447      .*     .      .      2.759267 

In  this  example,  the  sum  of  the  characteristics  is  8,  from 
which,  taking  10,  the  remainder  is  2. 

3.  Divide  37.149  by  523.76. 

log  37.149      .....     1.569947 
log  523.76         ar.  comp.  7.280867 

Quotient     .     .     0.0709273      .      .     .     2.850814 


252  '    BOOK    VIII SECTION    I. 

EXAMPLES   EMBRACING   MULTIPLICATION   AND   DIVISION. 

1    Multiply  3676  by  3278,  and  divide  the  product  by  2704* 
Common  way. 

log  3676 3.565376 

log  3278 3.515609 

sum 7.080985 

log  2704        3.432007 

•   Result   .     .     4456.34 3.648978 

By  Arithmetical  Complement. 
log  2704        arith.  comp.         6.567973 

log  3676 3.565376 

log  3278 3.515609 

Result   .     .     4456.34 3.648978 

2.  Multiply  600  by  600.11,   and  divide   the   product  by 
643.94. 

log  600       ......    2.778151 

log  600.11        2.778231 

5.556382 

log  643.94 2.808544 

log  559.55     .     .   . 2^747838 

3.  Multiply  780  by  1155.29,  and  divide   the   product   by 
218.64. 

log  218.64       arith.  comp.  7.660265 

log  780 2.892095 

log  1155.29 3.062692 

log  4121.47 3.615052 


LINES    AND    ANGLES.  253 

SECTION  II. 

LINES   AND    ANGLES. 

1.  The  radius  of  the  earth  being  very  large,  the  curvature 
may  'be  neglected,  when  lines  are  limited  to  small  portions  of 
the  surface. 

2.  The  line  of  water  level  is  called  a  horizontal  line  ;  and  the 
plane  of  water  level,  a  horizontal  plane. 

3.  All  lines  parallel  to  the  water  level  are  also  called  horl 
zontal  lines  ;  and  all  planes  parallel  to  the  plane  of  the  water 
level  are  called  horizontal  planes. 

4.  Lines  which  are  perpendicular  to  a  horizontal  plane,  are 
called  vertical  lines  ;  and  all  lines  which  are  inclined  to  it,  are 
called  oblique  lines. 

Thus,  AB  and  DC  are  hori- 
zontal lines;  BC  and  AD  are 
vertical  lines;  and  AC  and  BD 
are  oblique  lines. 

5.  The  horizontal  distance  be- 
tween two  points,  is  the  horizontal  line  intercepted  between 
the  two  vertical  lines  passing  through  those  points.  Thus, 
DC  or  AB  is  the  horizontal  distance  between  the  two  points 
A  and  C,  or  the  points  B  and  D. 

6.  A  horizontal  angle  is  one  whose  sides  are  horizontal;  its 
plane  is  also  horizontal. 

A  horizontal  angle  may  also  be  defined  to  be,  the  angle  in- 
cluded between  two  vertical  planes  passing  through  the  angular 
point,  and  the  two  objects  which  subtend  the  angle. 


254  BOOK    VIII SECTION    II. 

7.  A  vertical  angle  is  one,  the  plane  of  whose  sides  is  ver 
tical. 

8.  An  angle  of  elevation,  is  a  vertical  angle  having  one  of 
its  sides  horizontal,  and  the  inclined  side  above  the  horizontal 
side. 

Thus,  in  the  last  figure,  BA  C  is  the  angle  of  elevation  from 
A  to  C. 

9.  An  angle  of  depression,  is  a  vertical  angle  having  one  of 
its  sides  horizontal,  and  the  inclined  side  under  the  horizontal 
side.     Thus,  DC  A  is  the.  angle  of  depression  from  C  to  A. 

10.  An  oblique  angle  is  one,  the  plane  of  whose  sides  is 
oblique  to  a  horizontal  plane. 

11.  All  lines,  which  can  be  the  object  of  measurement,  must 
belong  to  one  of  the  classes  above  named,  viz. : 

1st.  Horizontal  lines : 
2d.  Vertical  lines : 
3d.  Oblique  lines. 

12.  All  the  angles  may  also  be  divided  into  three  classes, 
viz.  : 

1st.  Horizontal  angles : 

2d.  Vertical  angles ;  which  include  angles  of  elevation  and 
angles  of  depression  :  and 

3d.  Oblique  angles,  or  those  included  by  oblique  lines. 

MEASUREMENT    OF    DISTANCES. 

13.  Any  tape,  rod,  or  chain,  divided  into  equal  parts,  may 
be  used  as  a  measure ;  and  one  of  these  equal  parts  is  called 
the  unit  of  the  measure.  The  unit  of  a  measure  may  be  a 
foot,  a  yard,  a   od  or  any  other  ascertained  distance. 


MEASUREMENT    OF    DISTANCES.  255 

The  measure  in  general  use,  is  a  chain  of  four  rods  or  sixty- 
six  feet  in  length ;  it  is  called  Gunter's  chain,  from  the  name 
of  the  inventor.  This  chain  is  composed  of  100  links.  Every 
tenth  link  from  either  end,  is  marked  by  a  small  attached  brass 
plate,  which  is  notched,  to  designate  its  number  from  the  end. 
The  division  of  the  chain  into  100  equal  parts,  is  a  very  con- 
venient one,  since  the  divisions,  or  links,  are  decimals  of  the 
whole  chain,  and  in  the  calculations  may  be  treated  as  such. 

TABLE. 

1  chain  =  4  rods  =  66  feet  =  792  inches  =  100  links. 
Hence,  1  link  is  equal  to  7.92  inehes. 

80  chains  =  320  rods  =  1  mile. 
40  chains  =  \  mile. 
20  chains  ==  \  mile. 

14.  Besides  the  chain,  there  are  needed  for  measuring,  ten 
marking  pins,  which  should  be  of  iron,  each  about  ten  inches 
in  length  and  an  eighth  of  an  inch  in  thickness.  These  pins 
should  be  strung  upon  an  iron  ring,  and  this  ring  should  be 
attached  to  a  belt,  to  be  passed  over  the  right  shoulder,  sus- 
pending the  pins  at  the  left  side.  Two  staves  are  also  required. 
Each  of  these  should  be  about  six  feet  in  length,  and  have  a 
spike  in  the  lower  end  to  aid  in  holding  it  firmly,  and  a  hori- 
zontal strip  of  iron  to  prevent  the  chain  from  slipping  off; 
these  staves  are  to  be  passed  through  the  rings  at  the  ends  of 
the  chain. 

TO    MEASURE    A    HORIZONTAL    LINE. 

15.  At  the  point  where  the  measurement  is  to  be  begun, 
place,  in  a  vertical  position,  a  signal  staff,  having  a  small  flag 
attached  to  its  upper  extremity  ;  and  place  another  at  the  point 
where  the  measurement  is  to  be  terminated.  These  two  points 
are  generally  called  stations. 


256 


BOOK    VIII SECTION    II. 


Having  passed  the  staves  through  the  rings  of  the  chain,  let 
die  ten  marking  pins  and  one  end  of  the  chain  be  taken  by  the 
person  who  is  to  go  forward,  and  who  is  called  the  leader,  and 
let  him  plant  the  staff  as  nearly  as  possible  in  the  direction  of 
the  stations.  Then,  taking  the  staff  in  his  right  hand,  let  him 
stand  off  at  arm's  length,  so  that  the  person  at  the  other  end  of 
the  chain  can  align  it  exactly  with  the  stations :  when  the 
alignment  is  made,  let  the  chain  be  stretched  and  a  marking  pin 
placed ;  then  measure  a  second  chain  in  the  same  manner,  and 
so  on,  until  all  the  marking  pins  shall  have  been  placed.  When 
the  marking  pins  are  exhausted,  a  note  should  be  made,  that 
ten  chains  have  been  measured  ;  after  which,  the  marking  pins 
are  to  be  returned  to  the  leader,  and  the  measurement  con- 
tinued as  before,  until  the  whole  distance  is  passed  over.  It 
will  be  found  desirable  to  fasten  pieces  of  red  cloth  to  the 
heads  of  the  marking  pins,  that  they  may  be  more  readily 
found  in  thick  grass,  brushwood,  &c. 

Great  care  must  be  taken  to  keep  the  chain  horizontal,  and 
if  the  slope  of  the  ground  be  too  great  to  admit  of  measuring 
a  whole  chain  at  a  time,  a  part  of  a  chain  only  should  be 
measured :  the  sum  of  all  the  horizontal  lines  so  measured,  is 
evidently  the  horizontal  distance  between  the  stations. 

For  example,  in  measuring 
the  horizontal  distance  between 
A  and  C,  we  first  place  a  staff 
at  A  and  another  at- 6,  in  the 
direction  towards  C.  Then 
slide  up  the  chain  on  the  staff 
at  A  until  it  becomes  horizon- 
tal, and  note  the  distance  ab. 
Then  remove  the  staves  and  place  them  at  b  and  d ;    make  the 


MEASUREMENT  WITH  THE  TAPE  OR  CHAIN. 


257 


chain  horizontal,  and  note  the  distance  cd.  Measure  in  the 
same  manner  the  line  fC;  the  sum  of  the  horizontal  lines  ab, 
cd,  fC,  is  equal  to  AB,  the  horizontal  distance  between  A 
and  C. 


MEASUREMENT  WITH  THE  TAPE  OR  CHAIN. 

16.  We  now  propose  to  explain  the  best  methods  for  deter* 
mining  distances,  by  means  of  the  tape  or  chain  only. 

PROBLEM    I. 

To  trace,  on  the  ground,  the  direction  of  a  right  line,  that  shah 
be  perpendicular  at  a  given  point,  to  a  given  right  line. 

FIRST    METHOD. 

Let  BC  be  the  given  right  line,  and  A  the  given  point. 
Measure  from  A,  on  the  line  BC,  two 
equal  distances  AB,  AC,  one  on  each 
side  of  the  point  A.  Take  a  portion  of 
the  chain  or  tape,  greater  than  AB,  and 
place  one  extremity  at  B,  and  with  the 
other  trace  the  arc  of  a  circle  on  the  ground.  Then  remove  the 
end  which  was  at  B,  to  0,  and  trace  a  second  arc  intersecting 
the  former  at  D.  The  straight  line  drawn  through  D  and  A 
will  be  perpendicular  to  BC  at  A. 

SECOND    METHOD. 

Having  made  AB  =  AC,  take 
any  portion  of  the  tape  or  chain  con- 
siderably greater  than  the  distance 
between  B  and  C.  Mark  the  middle 
point  of  it.  and  fasten  its  two  extre- 
mities, the  one  at  B  and  the  other  at 
C.      Then,  taking  the  chain  by  the  middle  point,  stretch  it 


B 


258  BOOK    VIII SECTION    II. 

tightly  on  either  side  of  BC,  and  place  a  staff  at  D  or  Ei 
DAE  will  be  the  perpendicular  required. 

THIRD    METHOD. 


\s 


Let  AB  be  the  given  line,  and  C 
the  point  at  which  the  perpendicular  /]\ 

is  to  be  drawn.      From  the  point  C 

measure   a  distance    CA  equal  to  8.      A 

With    C  as   a   centre,  and   a   radius 

equal  to  6,  describe  an  arc  on  either 

side  of  AB:  then,  with  iasa  centre, 

and  a  radius  equal  to  10,  describe  a  second  arc  intersecting  at 

E,  the  one  before  described :   then  draw  the  line  EC,  and  it 

will  be  perpendicular  to  AB  at  C. 

Remark. — Any  three  lines,  having  the  ratio  of  6,  8,  and  10, 
form  a  right-angled  triangle,  of  which  the  side  corresponding  to 
10  is  the  hypothenuse. 

FOURTH    METHOD. 

Let  AD  be  the  given  right 
line,  and  D  the  point  at  which 
the  perpendicular  is  to  be  drawn. 
Take  any  distance  on  the  tape  or  f*  ^^ 
chain,  and  place  one  extremity  at 
D,  and  fasten  the  other   at   some  >\ 

point,  as  E,  between  the  two  lines  --•—-__—-' 

which  are  to  form  the  right  angle.  Place  a  staff  at  E.  Then, 
having  stationed  a  person  at  D,  remove  that  extremity  of  the 
chain  and  carry  it  round  until  it  ranges  on  the  line  DA  at  A. 
Place  a  staff  at  A  :  then  remove  the  end  of  the  chain  at  A,  and 


FRACTIONAL    PROBLEMS  259 

carry  it  round  until  it  falls  on  the  line  AE  at  F.  Then  place  a 
staff  at  F :  ADF  will  be  a  right  angle,  being  an  angle  in  a 
semicircle. 

PROBLEM    II. 

From  a  given  point  without  a  straight  line,  to  let  fall  a  perpen* 
dicular  on  the  line. 

Let  C  be  the  given  point,  and  AB  the  given  line. 
From    O   measure   a   line,   as  k 


CA,  to  any  point  of  the  line  AB. 

distance  as  AF,  and  at   F  erect         iti"'*"^ 


From    A,   measure    on    AB   any 


Jr^ 


G 


■tB 


FE  perpendicular  to  AB.  *  F  D 

Having  stationed  a  person  at  A,  measure  along  the  perpen- 
dicular FE  until  the  forward  staff  is  aligned  on  the  line  AC : 
then  measure  the  distance  AE.  Now,  by  similar  triangles,  we 
have, 

AE    :     AF    ::     AC    :     AD, 

in  which  all  the  terms  are  known,  except  AD,  which  may 
therefore,  be  found.  The  distance  AD  being  laid  off  from  A, 
the  point  D,  at  which  the  perpendicular  CD  meets  AB,  becomes 
known.  If  we  wish  the  length  of  the  perpendicular,  we  use  the 
proportion, 

AE    :     EF    :  :     AC    i     CD, 

in  which  all  the  terms  are  known,  excepting  CD :  therefore, 
CD  may  be  determined. 

EXAMPLES. 

1.  Let  us  suppose  AE  =  334  yards;  EF  =  247  yards; 
and  AC  =  560  :  find  the  value  of  CD  by  logarithms. 


260  BOOK    VIII SECTION    II. 

log  AE  =  334       arith.  comp.  7.476254 

log.  EF=  247       .....  2.392697 

log  AC  =560        2.748188 

log  CD  =  414.1324         .     .     .  2.617139 


2.  Let  us  suppose  AE  =  668  feet ;   EF  =  494  feet ,   and 
AC  =  1120  feet :    what  will  be  the  length  of  CD  ? 

Ans.  828.2648/*. 

3.  Let  us  suppose  AE  =  1002  rods ;    EF  =  741  rods  ;  and 
AC  =  1680  rods  :  what  will  be  the  length  of  CD? 

Ans.  1242.3972  rods. 

PROBLEM    III. 

To  determine  the  horizontal  distance  from   a  given  point  to 
an  inaccessible  object. 

FIRST    METHOD. 

Let  A  be  an  inaccessible  object,  and  E  the  point  from  which 
the  distance  is  to  be  measured. 

At  E  lay  off  the  right  angle 
AED,  and  measure,  in  the  direc- 
tion ED,  any  convenient  distance 

to  D,  and  place  a  staff  at  D.    Then  -^^*^      ~Z 

measure  from  E   directly  towards  h  ,<"' 

the  object  A,  a  distance  EB  of  a  g        ^\ 

convenient  length,  and  at  B  lay  off  \.S       \ 

a   line  BC  perpendicular    to  EA.  &            ™ 


Measure  along  the  line  BC,  until  a  person  at  D  shall  range 
the  forward  staff  on  the  line  DA.     Now,  DF  is  known,  being 


PRACTICAL  PROBLEMS.  261 

equal  to  the  difference  between  the  two  measured  lines  DE  and 
CB.     Hence,  by  similar  triangles, 

DF    :     FC    :  :     DE    :     FA, 
in  which  proportion  all  the  terms  are  known,  except  the  fourth, 
which  may,  therefore,  be  found. 

EXAMPLES. 

1.  Suppose  DF—  269   yards;     FC  =  150    yards;     and 
DE  =412  yards  :  find  the  length  of  EA  by  logarithms. 

logZ>i^.  .269  arith.  comp.  7.570248 

log  FC .  .  150 2.176091 

log  DE .  .  412 .  2.614897 

log  EA  .  .  229.740  nearly       ...  2.361236 

2.  Suppose    DF  =  807     feet ;     FC  =  450     feet ;     and 
DE  =  1236  feet :  what  is  the  length  of  EA  ? 

Ans.  689.22  ft. 

3.  Suppose  DF  =  538  rods ;  FC  =  300  rods ;   DE  =  824 

rods :  what  is  the  length  of  EA  ? 

Ans.  459.48  rods. 

SECOND   METHOD. 

At  the  point  E  lay  off  EB 
perpendicular  to  the  line  EA, 
and  measure  along  it  any  con- 
venient distance,  as  EB.  /' 

At  B  lay  off  the  right  angle       B C\/'' 

EBD,  and    measure    any  dis- 
tance in  the  direction  BD.    Let 
a  person  at  D  align  a  staff  on        %> 
DA,  while    a   second    person    at   B   aligns    it   on    BE:   the 


E 


262  BOOK    VIII SECTION    II. 

staff  will    thus    be  fixed    at    C.      Then    measure   the   dis- 
tance BO. 

The  two  triangles  BOD  and  OAE  being  similar,  we  have, 
BO    :     BD     ::     OE    :     EA, 
in  which  all  the  terms  are  known,  except  the  fourth,  which  may, 
therefore,  be  found. 

EXAMPLES. 

1.  Suppose   BO  =  112    yards;     BD  =  89    yards;     and 
OE  =  224  yards  :  find  the  distance  EA  by  logarithms. 

log  BO.  .112  arith.  comp.  7.950782 

log  BD  ,  .  89  ........  1.949390 

log  OE .  .  224 2.350248 

log  EA  .  .  178 2.250422- 

2.  Suppose    BO  =   336    feet;     BD  =  267    feet;     and 
OE  =  672  feet :  what  is  the  distance  EA  f 

Ans.  534  ft. 

3.  Suppose    BO  =  224    rods ;     BD  =  178    rods ;      and 
OE  =  448  rods  :  what  is  the  distance  EA  f 

Ans.  356  rods. 

PROBLEM    IV. 

To  find  the  altitude  of  an  object,  when  the  distance  to  the 
vertical  line  parsing  through  the  top  can  be  measured. 

Let    CD  be   the   altitude    required,  and  AC  the  measured 
distance. 

From  A,  measure   on   the   line   AC,   any  convenient   dis- 


PRACTICAL    PROBLEMS. 


263 


tance  AB,  and  place  a 
staff  vertically  at  B. 
Then  placing  the  eye 
at  A,  sight  to  the  ob- 
ject D,  and  let  the 
point,  at  which  the 
line  AD  cuts  the  staff 
BE,  be  marked.     Measure  the  distance  BE  on  the  staff;    then, 

AB     :     BE    :  :     AC    :      CD,  • 

whence  CD  becomes  known. 

If  the  line  A  C  cannot  be  measured,  on  account  of  inter- 
vening objects,  it  may  be  determined  by  calculation,  as  in  the 
last  problem,  and  then,  having  found  the  horizontal  distance, 
the  vertical  line  is  readily  determined,  as  before. 

EXAMPLES. 

1.  Suppose  AB  --=  36  feet ;  BE  =  12  feet ;  and  AC  =  354 
feet :  what  is  the  height  CD  ? 

36     :     12     :  :     354     :     118  =  CD. 

BY    LOGARITHMS. 

log  AB  .  .36  arith.  comp.  8.443697 

log  BE.  .  12        1.079181 

log  A  C.  .354 2.549003 

log  CD..  US 2^071881 

2.  Suppose  AB  =  108  feet ;  BE  =  36  feet ;  and  AC  =  1062 
feet :  what  is  the  height  CD  ? 

Ane,  354//. 

3.  If  AB  =  72  feet ;  BE  =  24feet ;  and  AC  =  708  feet! 
what  is  the  height  CD  ? 

Avs.  236//. 


264  BOOK    VIII. SECTION    II. 

PROBLEM   V. 

To  measure  heights  by  means  of  shadows. 

The  length  of  the  shadow  of  a  pole  standing  vertically  on  a 
horizontal  plane,  is  to  the  length  of  the  shadow  of  any  vertical 
object,  as  the  length  of  the  pole  to  the  height  of  the  object. 

Let  BO  be  any  vertical  ob- 
ject standing  on  a  horizontal 
plane,  and  AB  the  length  of 
its  shadow. 

Having  measured  and  erect- 
ed a  vertical  staff,  and  found  ~B 
the  length  of  its  shadow,  let  P  denote  the  length  of  the  staff, 
and  S  the  length  of  its  shadow :  then, 

S    :     AB    :  :     P    :     BO; 
hence,  BO  =  ~ 

EXAMPLES. 

1.  If  a  pole  10  feet  high  cast  a  shadow  14  feet  long :  re- 
quired the  height  of  an  object  which  casts  a  shadow  65  feet 
long. 

^=^^  =  ^  =  46.43 
&  14 

2.  If  a  pole  20  feet  long  casts  a  shadow  of  28  feet :  required 
the  height  of  an  object  which  casts  a  shadow  of  130  feet. 

Ans.  92.86. 


PRACTICAL    PROBLEMS. 


265 


GEOMETRICAL    SQUARE. 

A  geometrical  square  is  an  instru- 
ment, ABCD,  in  the  form  of  a 
square,  with  two  sights,  A  and  B, 
on  one  edge,  and  having  the  two 
edges,  CD,  AC,  divided,  each  into 
100  equal  parts,  the  graduation  be- 
ginning at  A  and  C,  and  ending  at  J). 

The  edge  AB  is  called  the  directive  edge,  CD  the  parallel  edge, 
and  AC  the  perpendicular  edge.  A  plummet  is  attached  to 
the  instrument  by  means  of  a  thread  suspended  from  the  ver- 
tex of  the  angle  B. 

When  the  instrument  is  used,  the  line  of  sight  AB  is  directed 
to  the  top  of  the  object  whose  altitude  is  to  be  measured.  We 
then  note  the  distance  cut  off  by  the  plumb  line,  either  on  the 
parallel  edge  CD,  or  on  the  perpendicular  edge  AD.  The 
number  denoting  such  distance,  together  with  the  length  of  the 
base  line  passing  through  the  foot  of  the  object,  will  enable  us 
to  find  the  height. 

PROBLEM    VI. 

To  measure  the  height  of  an  object  by  means  of  the  geometrical 

square. 
Case  I. — When  the  intercepted  part  is  on  the  parallel  side. 

Let   LH  be   a    light-  H 

house,  whose  height  is 
required.  Direct  the 
line  of  sight  AB  to  the 
top,  H,  of  the  object ; 
then  observe  the  num- 
ber of  parts  in   CE,  and  measure  the  base  FL. 

12 


266 


BOOK    VIII. SECTION    II. 


The  triangles  BCE  and  FLU  are  similar     hence, 

BC    :     CE    :  :     FL     :     LH ; 
CExFL 


hence, 


LH=z 


BC 


EXAMPLES. 

1.  What  is  the  height  when  CE=  84  and  FL  —  120  feet  9 
Since  BC  =  100,  we  have, 

TTT     84  X  120 

LH—  —^r —  =  100.8  feet. 

2.  What  is  the  height  when  CE  =  75  and  FL  =  125  feet  ? 

Ans.  93.75/*. 

3.  What  is  the  height  when  CE  —  45  and  FL  =  240  yards  ? 

-4ns.  108  yds. 


Case  II. — When  the  intercepted  part  is  on  the  perpendicular 
side. 


Let  PN  be  the  object,  and  MN  the 
measured  base  :  then,  since  the  tri- 
angles ABE  and  MNP  are  similar, 


AE 
whence, 


AB 


MN 


NP; 


NP 


AB  xMN 
AE 


EXAMPLES. 


1.  What  is  the  height  when  AE  =  40  and  MN  =  250  feet? 

Ans.  626  ft. 

9.  WThat  is   the   height  when  AE—6h    and   JfiV^  260 
yards  1  Ans.  400  yds. 


PRACTICAL    PROBLEMS.  267 

PROBLEM    VII. 

To  measure  the  height  of  an  object  by  means  of  a  plane  mirror. 

Let  DE  be  the  object  whose  altitude  is  required,  and  A  CD 
a  line  in  the  same  horizontal  plane  with  the  base. 

Place  a  mirror, 
A,  horizontally  on 
the  plane,  and  in 
such  a  position 
that,  from  the  top 
of  a  vertical  staff 
BC,  the  image,  E, 

of  the  top  of  the  W 

building  will  be  seen  in  the  mirror,  in  the  direction  BAE. 
Measure  the  distances  AC,  BC,  and  AD :  then, 

AC    :     CB    :  :     AD     :     DE'  or  DE; 
ADxBC 


hence,  DE  = 


AC 


EXAMPLES. 

1.  Given  AC  =  10,  BC  =  8,  and  AD  =  84  feet,  to  find  the 
height  DE. 

AC  10 

2.  Given  A  C  =  12,  BC  =  9,  and  AD  =  125  feet :  required 

the  height  DE. 

Ans.  DE=  93.75  //. 

3.  Given  AC—  18,  BC  —  13.5,  and  AD  =  187.5  feet:  re- 
quired the  height  DE. 

Ans.  DE-  140//. 


2fi8  BOOK    VIII. SECTION    If. 

PROBLEM    VIII. 

To  determine  distances  by  means  of  sound. 

The  velocity  or  rate  of  travel,  with  which  sound  passes, 
through  the  atmosphere  has  been  determined  with  considerable 
precision,  at  least,  to  within  about  a  two-hundredth  part  of  the 
distance  passed  over  in  a  given  or  known  time.  Hence,  this 
velocity  can  be  employed  to  determine  the  distance  at  which 
the  cause  of  any  sound — as  the  report  of  a  cannon  or  a  peal  of 
thunder — has  happened,  when  the  time  elapsed  between  the 
flash  of  the  powder,  or  of  the  lightning,  and  the  perception  of 
the  sound  is  known. 

It  has  been  found,  by  observation,  that  the  velocity  of  sound 
is  about  1125  feet  in  a  second  of  time;  hence,  if  the  time  be 
estimated  in  seconds,  and  denoted  by  T,  and  the  required  dis- 
tance by  J),  we  shall  have, 

D  =  1125  x  T. 

EXAMPLES. 

1.  What  is  the  distance,  when  the  time  is  5  seconds'? 

D  =  1125  X  T-  1125  x  5  -^  5625  feet. 

2.  What  is  the  distance  of  a  ship,  having  observed  that  the 

report  of  a  gun  fired  on  board  it,  was  heard  10  seconds  after 

the  flash  was  seen  1 

Ans.  2  miles  69/£. 

3.  Find  the  distance  of  a  thunder   cloud,  when  the  time 

elapsed  between  the  flash  of  lightning  and  the  thunder  is  0 

seconds. 

Ans.  1  mile  490  yds. 


PRACTICAL    PROBLEMS.  269 

LEVELLING. 

A  line  of  true  level  is  such,  that  all  points  in  it  are  equally 
distant  from  the  centre  of  the  earth. 

A  line  of  apparent  level,  at  any  place,  is  a  horizontal  line 
passing  through  that  place. 

Let  LT  be  an  arc  of  the  earth's  sur- 

L  P 

face,  and  LTNM  a  portion  of  the  earth, 

and  LP  a  line   drawn   tangent   to   the 

arc  at  the  point  L.     Then,   L  and   P 

are  in  the  same  apparent  level,  when  P  is  seen  in  a  horizontal 

line  from  L ;  also,  L  and  T  are  in  the  same  true  level,  and 

PT  is  the  difference  between  the  true  and  apparent  level  of  the 

points  L  and  P.     The  horizontal  line  through  L  is  determined 

by  means  of  a  spirit  level  placed  at  L. 

The  spirit  level,  SL,  consists  of  a  glass  tube  nearly  filled 

with  spirit  of  wine,  and  enclosed  in  a  brass  tube,  except  the 

upper  part.     It  is  sometimes  placed  parallel  to  the  axis  of  a 

telescope,  and  when  brought  to  a  level,  a  point  at  a  distance  is 

seen  on  the  same  level  with 

the  axis  of  the   telescope,     £m> 

by  looking  through  it  to  a  ^ 

pole  or  other  object  at  a     \\  ^ 

distance,  and  marking  the        F  7^  G~ 

point  on  it  that  appears  to 


^ 


I 


coincide  with  the  intersec- 
tion of  two  very  fine  wires  that  cross  each  other  within  the 
telescope. 

The  spirit  level  is  also  sometimes  attached  to  a  bar  of  brass, 
FG,  with  two  upright  pieces,  FE,  GO,  and  small  openings  or 
sights  at  E  and  0  so  placed  as  to  be  on  a  horizontal  line  when 


270  BOOK    VIII. SECTION    III. 

the  level  SL  is  horizontal,  which  is  always  the  case  when  the 
air  bubble  at  B  is  at  the  middle  point. 

PROBLEM    IX. 

To  find  the  difference  between  the  true  and  apparent  level  for 
any  given  distance. 

The  difference  FT  between  the  true  and  apparent  level,  for 
any  distance  LT  (see  fig.,  p.  269),  is  found  to  be  proportional 
to  the  square  of  the  distance  LT.  For  1  mile  this  difference 
is  found  to  be  8  inches ;  for  2  miles  it  will  be  8  inches  multi- 
plied by  the  square  of  2,  which  is  4 ;  for  3  miles,  8  inches 
multiplied  by  the  square  of  3,  which  is  9 ;  and  similarly  for 
any  other  distance. 

EXAMPLES. 

1.  What  is  the  difference  between  true  and  apparent  level, 
at  a  distance  of  4  miles  ? 

42  =  16,  and  16  X  8  inches  =  128  inches  =  10J  feet. 

2.  What  is  the  difference  between  true  and  apparent  level, 
at  a  distance  of  2022  feet? 

First,  find  what  part  the  given  distance  is  of  one  mile. 

OAOO  _  2 

=^s  =  0383  mile:  hence>  '383    X  8  =  1-l74c  inches. 

3.  Required  the  difference  between  true  and  apparent  level, 
at  the  distance  of  2J  miles.  Ans.  4  ft.  2  in. 

4.  If  at  a  point  on  the  surface  of  a  canal  it  is  found  that  for 
a  distance  of  3  J  miles  the  surface  of  the  earth  is  on  an  appa- 
rent level  with  it,  required  the  depth  of  the  surface  of  the 
canal  below  the  surface  of  the  earth,  at  that  distance  1 

Ans,  Sft.  2  in. 


STRENGTH    OF    MATERIALS.  271 

SECTION     III. 
STRENGTH    OF    MATERIALS. 

1.  In  hrvestigating  the  strength  of  beams  of  various  mate- 
rials, as  timber  and  metals,  they  are  conceived  to  be  composed 
of  equal  elastic  fibres  disposed  in  the  direction  of  their  length. 
The  distention  is  considered,  within  certain  narrow  limits,  to 
be  proportional  to  the  distending  force.  When  a  beam  is  sub- 
jected to  a  strain  exceeding  its  elastic  force,  it  takes  a  set ;  that 
is,  a  permanent  alteration  of  form  ;  and  a  force  just  not  suffi- 
cient to  produce  this  effect,  is  considered  to  measure  the 
strength  of  the  beam.  The  rules  deduced  by  the  aid  of  theory 
and  experiment  afford,  in  many  important  cases,  a  tolerably 
near  approximation  to  fact,  considering  the  variety  of  strength 
of  materials  of  the  same  kind. 

2.  The  measure  of  the  absolute  strength  or  direct  cohesion  of 
any  material,  is  the  greatest  weight  that  a  prism,  one  inch 
square,  is  capable  of  supporting,  acting  in  the  direction  of  its 
length. 

The  weight  thus  supported  by  any  body  will  evidently  be 
proportional  to  its  transverse  section. 

3.  When  a  beam,  as 
2?iV,  with  one  end 
fixed,  is  strained  by  a 
weight,  W,  suspended 
from  any  part  of  it, 
the  moment  of  the 
greatest  weight  it  can  support  (that  is,  the  product  of  the 
greatest  weight  supported  into  the  length  of  the  lever,  BM', 
with  which  it  acts)  is  the  measure  of  its  transverse  or  relative 
strength. 


272  BOOK    VIII. SECTION    III. 

The  effect  of  the  straining  weight  is  to  distend  the  fibres  on 
the  upper  part  of  the  beam,  and  to  compress  those  in  the  lower 
part ;  and  the  length  of  the  fibres  in  some  intermediate  posi- 
tion, as  AX,  will  not  be  altered  either  by  distention  or  com- 
pression, so  that  AX  will  be  what  is  called  the  neutral  axis ; 
and  the  line  passing  through  the  neutral  point  A,  about  which 
the  motion  takes  place,  is  called  the  neutral  axis  of  rotation. 
When  the  beam  is  suffering  the  greatest  strain  it  can  bear,  the 
fibres  at  B  are  exerting  their  absolute  strength,  and  the  interme- 
diate fibres  between  B  and  A  suffer  distentions  proportional  to 
their  distances  from  A.  The  fibres  on  the  lower  side  are  com- 
pressed, the  quantity  of  compression  being  proportional  to 
their  distances  from  A.  If  BB'  represent  the  distention  of 
the  fibre  BM,  PP'  will  be  the  distention  of  a  fibre  at  the  dis- 
tance AP  from  A ;  DD'  will  be  the  compression  of  the  fibre 
DN,  and  CC  that  of  a  fibre  at  the  distance  A  0  from  A.  The 
distention  and  compression  are  by  some  considered,  when  not 
great,  to  be  equal ;  although,  in  the  case  of  unseasoned  timber, 
the  resistance  to  compression  exceeds  that  to  distention. 

The  weight,  W,  supported,  will  be  inversely  as  the  length 
of  the  beam  AX;  for  it  is  the  arm  of  the  lever  at  which  W 
acts.  It  is  also  evident  that  the  weight  will  be  proportional 
to  the  breadth;  for  if  another  beam,  equal  to  BN,  were  joined 
to  its  side,  the  double  beam  would  support  a  weight  =  2  W. 
The  weight  will  also  be  proportional  to  the  square  of  the  depth. 
This  is  proved  in  treatises  of  theoretical  mechanics,  and  also 
in  practical  treatise*.*  Hence,  the  transverse  strength  is 
proportional  to  Wbd2 

T' 

*  See  Barlow's  Essay  on  the  Strength  of  Timber,  and  Tredgold's  E«?ay 
on  the  Strength  of  Cast-Iroa 


STRENGTH    OF    MATERIALS.  273 

4.  The  quantity  by  which  a  beam  is  bent  from  its  position 
of  rest  by  a  transversely  straining  force,  is  called  the  deflexion 
as  MM1. 

5.  When  a  pressure  acts  against  a  beam  in  the  direction  of 
its  length,  as  in  the  case  of  pillars  or  posts,  it  is  called  a  crush 
ing  or  compressive  force. 

6.  A  force  acting  in  such  a  manner  as  to  twist  or  wrench  a 
beam,  as  when  it  acts  at  the  free  extremity  of  an  arm  of  a 
lever,  one  end  of  which  is  fixed  in  the  beam  in  a  direction  per- 
pendicular to  it,  as  in  the  case  of  the  axles  of  wheels,  or  the 
screw  of  a  press,  is  called  a  force  of  torsion  ;  and  the  angle 
through  which  it  is  twisted  is  called  the  angle  of  flexure. 

7.  The  strains  to  which  a  bar  of  timber,  metal,  or  other  hard 
materials,  may  be  subjected,  are  reducible  to  the  four  already 
explained,  namely,  a  direct  strain,  a  transverse  strain,  a  force  of 
compression,  and  a  force  of  torsion. 

8.  The  modulus  of  elasticity  is  that  length  of  a  prismatic 
beam,  whose  weight  would  be  capable  of  distending  any  por- 
tion of  it  to  double  its  length,  were  it  capable  of  such  disten- 
tion supposed  to  be  uniform. 

Hence,  any  small  distention  of  a  beam  is  to  its  length,  as  the 
distending  weight  to  the  weight  of  the  modulus  of  elasticity. 

9.  The  following  table  contains  the  data  for  determining  the 
strength  and  flexibility  of  materials.  The  column  G  contains 
the  specific  gravity ;  C  the  absolute  strength  or  direct  cohe- 
sion ;  S  the  constant  for  transverse  strain  ;  E  the  constant  for 
deflexion  ;  U  that  for  ultimate  deflexion  ;  and  M  the  modulus 
of  elasticity. 

12* 


274 


BOOK    VIII. SECTION    III. 


Material. 

6 

c 

s 

E 

u 

M 

Ash,   - 

760 

17000 

2030 

6580000 

395 

4988000 

Beech, 

700 

11500 

1560 

5417000 

615 

4457000 

Birch,  Common,  - 

700 

1900 

6570000 

5406000 

"   American, 

150 

1500 

5700000 

3388000 

Elm,  - 

540 

5780 

1030 

2803000 

509 

3007000 

Fir,  Mar  Forest, 

700 

12000 

1140 

3400000 

588 

2797000 

"  Riga,   - 

750 

12600 

1130 

5314000 

588 

4080000 

"  New  England, 

550 

12000 

1100 

5967000 

757 

6249000 

Larch,  Scotch, 

540 

7000 

1120 

4200000 

411 

4480000 

Oak,  English,  j  [™m 

900 

9000 

1200 

3490000 

598 

2872000 

15000 

2260 

7000000 

4702000 

"  Adriatic, 

990 

14000 

1380 

3880000 

610 

2257000 

*     Canadian, 

872 

12000 

1760 

8950000 

588 

5674000 

"  Dantzic, 

156 

14500 

1450 

4760000 

724 

3607000 

Pine,  Red, 

660 

10000 

1340 

7360000 

605 

6423000 

"  Pitch,  • 

660 

10500 

1630 

5000000 

588 

4364000 

Poon, 

600 

14000 

2200 

6760000 

596 

6488000 

Spar,  Norway, 

577 

12000 

1470 

5830000 

648 

5789000 

Teak, 

750 

15000 

2460 

9660000 

818 

7417000 

Iron,  Cast,  j*om  ' 

7200 
7760 

16300 
36000 

8100 

69120000 

5530000 

■  Malleable,  • 

60000 

9000 

91440000 

6770000 

"  Wire, 

80000 

PROBLEM    I. 


10.    To  find  the  absolute  strength  of  any  piece  of  any  material 
of  given  dimensions. 


RULE. 

Find  the  area  of  the  transverse  section  in  square  inches,  and 
multiply  it  by  the  value  of  (7  for  the  given  material  in  the  pre- 
ceding table,  and  the  product  will  be  the  required  strength. 

Let  a  --  the  area  of  the  transverse  section  in  inches, 
s  =  the  strength  in  pounds, 
then      s  =  aC. 

When  s  is  given,  and  a  is  required,  a  =  — ,  and  when  a  is  a 
square,  whose  side  is  6,  then  6  ==  y'aT 


STRENGTH    OF    MATERIALS.  275 

When  a  is  a  rectangle,  whose  breadth  is  b,  and  depth  d,  and 
if  6  is  given,  then  d  —  — - ;  and  if  d  is  given,  b  =—.     If  a  is 


a  circle,  whose  diameter  is  d,  then  d  =  \t  ~ftP,  v 

The  constant,  C,  could  also  be  calculated,  if  a  and  s  were 
found  by  experiment,  for  C  =  — . 

When  the  specific  gravity  differs  from  that  in  the  table,  then 
the  tabular  specific  gravity  is  to  that  given,  as  the  strength 
found  by  the  above  rule  to  the  required  strength. 

Let  g  =  the  given  specific  gravity, 

then,  in  this  case,  s  =  ~^-. 

EXAMPLES. 

1.  What  weight  will  be  necessary  to  tear  asunder  a  rectan 
gular  piece  of  beech,  whose  breadth  (b)  and  depth  (d)  are  res- 
pectively 6  and  3  inches  ? 

Here  a  =  bd  =6x3=  18  inches ; 
hence,  s  =  aC  =  18  x  11500  =  207000  lbs. 

But  if  the  specific  gravity  were  different  from  that  in  the 
table,  as  705,  then  s  =  ^  =  ^5  x  207000  =  208478. 

2.  What  weight  will  be  sufficient  to  tear  asunder  a  square 

piece  of  ash,  the  side  of  which  is  3  inches  ? 

Ans.  153.000  lbs. 

3.  What  must  be  the  weight  necessary  to  tear  a  cylinder 
of  cast-iron  2  inches  diameter  of  mean  strength,  for  which 
C  =  48000  ?  Ans.  150707  lbs. 


276  BOOK    VIII. SECTION    III. 

IROBLEM    II. 

11.    To  find  the  deflexion  cf  a  beam  fixed  at  one  end  and  loaded 
at  the  other. 

RULE. 

Find  the  continued  product  of  the  tabular  value  of  E^  the 
breadth  of  the  beam,  and  the  cube  of  its  depth,  both  in  inches  •, 
find  also  the  continued  product  of  32,  the  given  weight  in 
pounds,  and  the  cube  of  the  length  in  inches  ;  divide  the  latter 
product  by  the  former,  and  the  quotient  will  be  the  deflexion 
in  inches. 

When  the  weight  is  uniformly  distributed  along  the  beam, 
the  same  rule  applies,  except  that  the  factor  32  must  be 
changed  to  12. 

The  rule  applies  only  to  cases  of  small  deflexion.    . 

When  the  specific  gravity  differs  from  that  given  in  the  table, 
a  proportion  must  be  made,  as  in  the  last  problem — namely, 
the  tabular  specific  gravity  is  to  that  given,  as  the  result  found 
by  the  preceding  rule  to  the  true  deflexion. 

Let  b,  I,  d,  W  =  the  breadth,  length,  and  depth  of  the  beam 
and  the  weight,  respectively,  and  D  =  the  deflexion,  then 

and  when  the  load  is  uniformly  distributed, 
D  =  12TF73  -r-  McP. 

EXAMPLES. 

1.  How  much  will  a  batten  of  larch,  5  feet  long,  1^  inches 
broad,  and  2^  deep,  be  deflected  by  a  weight  of  15  lbs.  sus- 
pended  from  its  free  extremity  % 


STRENGTH    OF    MATERIALS.  277 

D  =  32WP+Ebd?=  32  x  15  X  603  -7-  4200000  X  1}  X  (f)2 

32  x  15  x  216000  x  2  X  8  _  16  X  10  X  216  X  64 

""         4200000  x  3  x  125         ~~     2100  x  1  X  1000 

16  X  1  X  72  X  64       -■  —  ... 

= — — — — =r  1.05  inches. 

700  X  100 

Formulas  may  easily  be  obtained  from  the  rule  to  find  any 
one  of  the  quantities  /,  b,  d,  W,  2>,  or  E,  when  the  rest  are 
given ;  thus, 

TTr      DEbd3   „      DEbd3  _       32  WP  ^      32  WP 

W   —  . /o   —  A  —  Ho  —   

~~    32/3   '      ~  32  IT '  DEd?  '      ~    DM  ' 

S2PW 


When   the  beam  is  square  b  =  d,  and  then  54 
The  constant,  E,  could  be  also  found,  for  E  = 


BE 

Z2WP 


Dbd 

2.  A  spar  of  Mar  Forest  fir  is  10  feet  long,  2  inches  broad, 
and  3  deep,  what  will  be  its  deflexion  if  a  weight  of  10  lbs.  be 
suspended  from  its  free  end  ?  Am.  3  in. 

3.  What  will  be  the  deflexion  of  the  same  spar,  if  it  is  sup- 
ported by  a  spur  at  4  feet  distance  from  the  wall1? 

4  Ans.  .65  in. 

PROBLEM    III. 

12.    To  find  the  deflexion  of  a  beam  supported  at  both  ends,  and 
loaded  with  a  weight  at  the  middle. 

RULE. 

Find  the  continued  product  of  the  tabular  value  of  E,  the 
breadth,  and  cube  of  the  depth,  both  in  inches ;  find  also  the 
product  of  the  cube  of  the  length  in  inches  into  the  given 
weight  in  pounds ;  then  divide  the  latter  product  by  the  for. 
mer,  and  the  quotient  wil   be  the  deflexion  in  inches. 


278  BOOK    VIII. SECTION    III. 

13.  When  the  beam  is  fixed  at  both  ends,  the  deflexion, 
with  an  equal  weight,  will  be  only  §  of  the  result  found  by  the 
above  rule. 

D=  WP  +  Md3-, 

and  from  this  formula  are  derived  the  following : 

It  is  evident,  from  the  above  rule,  that  the  deflexion  in  the 
former  problem  for  the  same  beam  and  weight  is  32  times 
greater  than  in  this  problem. 

EXAMPLES. 

1.  Find  the  deflexion  of  a  beam  of  the  best  English  oak,  its 
length  being  20  feet,  breadth  4  inches,  and  depth  5  inches, 
when  loaded  with  a  weight  of  1000  lbs. 

Wl3       1000  x  203  X  123       1  x  8000  x  432 


Z>  = 


Ebd3      7000000  x  4  x  53  ~  7000  X  1  X  125 

1  X  64  x  432      27648       0  nc  .    _. 

3'9&  inches. 


7  X  1  X  1000        7000 

2.  A  joist  of  American  birch,  20  feet  long,  3  inches  thick, 
and  8  deep,  is  loaded  with  a  weight  of  3000  lbs. :  what  is  the 
depression?  Ans.  4.7  in. 

3.  A  beam  of  iron  30  feet  long,  3  inches  thick,  and  9  inches 
deep,  is  fixed  at  both  ends,  and  loaded  with  3  tons  :  what  is  the 
deflexion1?  A ns.  1.38  in. 

14.  When  the  beam  is  uniformly  loaded,  and  supported  at 
both  ends,  the  deflexion  will  be  only  f  of  that  found  by  the 
preceding  problem. 

15.  When  the  beam  is  uniformly  loaded  and  fixed  at  both 
ends,  the  deflexion  will  be  only  fa  of  that  found  by  the  prece- 


STRENGTH    OF    MATERIALS.  279 

ding  problem,  or  §  of  that  found  by  the  rule  in  the  preceding 
article. 

16.  The  deflexion  of  a  beam,  fixed  at  both  ends,  is  only  § 
of  the  deflexion  when  its  ends  are  only  supported,  whether  it 
be  loaded  by  a  weight  at  the  middle,  or  by  a  weight  uniformly 
distributed  over  it. 

The  examples  in  the  preceding  problem  are  sufficient  to 
illustrate  the  rules  in  articles  14  and  15. 

6WP 


The  formula  for  the  case  in  Art.  14  is  D  = 
and  that  for  the  case  in  Art.  15  is  D 


8  Ebd? 

12  Ebd? 

And  from  these  the  formulas  for  b,  d,  I,  and  W,  are  easily  ob- 
tained. 

PROBLEM    IV. 

17.    To  find  the  ultimate  deflexion  of  a  beam  before  rupture^ 
when  supported  at  both  ends. 

RULE. 

Multiply  the  tabular  value  of  U  by  the  depth  of  the  beam  in 
inches,  and  divide  the  square  of  the  length  also  in  inches  by 
that  product,  and  the  quotient  will  be  the  ultimate  deflexion. 

D  =  P--dU. 
Hence,  I2  =  dDU,  and  d  =  P  <r  DU. 

18.  The  same  rule  applies  when  the  beam  is  fixed  at  both 
ends. 

19.  When  the  piece  is  fixed  only  at  one  end,  the  deflexion 
will  be  8  times  greater  than  that  found  by  the  above  rule,  or 
D  ----  8  I2  ~  dU. 


280  BOOK    VIII. SECTION    III, 

EXAMPLES. 

1.  Find  the  ultimate  deflexion  of  a  2-inch  plank  of  Dantzic 
oak,  which  is  25  feet  long. 

D  =  P  +  dU  =  3002-r2  X  724  =  ^22 

1448 

=  62.16  =  5  feet  2  inches. 

2.  Find  the  ultimate  deflexion  of  an  ash  plank  4  inches 
thick  and  40  feet  long.  Ans.  145.8  in. 

3.  A  spar  of  ash,  2  inches  deep  and  6  feet  long,  is  fixed  at 
one  end,  and  is  broken  by  a  weight  applied  at  the  free  end  : 
what  was  the  ultimate  deflexion  ?  Ans.  52.5  in. 

problem  v. 

20.     To  find  the  ultimate  transverse  strength  of  a  rectangular 

beam,  fixed  at  one  end,  and  loaded  at  the  other. 

RULE. 

Find  the  continued  product  of  the  tabular  value  of  S,  the 
breadth  and  square  of  the  depth,  both  in  inches,  and  divide  this 
product  by  the  length  in  inches,  and  the  quotient  will  be  the 
weight  in  pounds. 

Or,  W=bd*S+l. 

7      od2S  -_~    IW V-  - ■'  ■        IW 
Hence,  J=  _,  J=  _   and  eP  =-^. 

21.  When  the  beam  is  uniformly  loaded,  the  weight  will  be 
twice  as  great  as  that  found  by  this  rule. 

Besides  this  rule,  another,  which  is  more  accurate,  is  given 
in  Barlow's  Essay,  which,  in  the  examples  given  by  him,  leads 
to  a  result  from  T^  to  fe  greater  than  that  obtained  by  the 
above  rule.  The  latter  rule,  however,  is  more  simple,  and 
therefore  more  convenient  in  practice,  and  for  a  permanent 
load,  only  f-  of  that  obtained  by  this  rule  is  taken. 


STRENGTH    OF    MATERIALS.  281 

EXAMPLES. 

1.  Find  the  ultimate  transverse  strength  of  a  malleable  bar 
of  iron,  10  feet  long,  2  inches  thick,  and  3  deep. 

w=  bd*s+  i  =  2x&x  90oo~i20  =  ?2Ljy£°?22 

=  1350  lbs. 

2.  What  weight  will  break  a  spar  of  New  England  tir,  its 
breadth  being  2  inches,  depth  3  inches,  and  length  5  feet  ? 

Ans.  330  lbs. 

3.  What  weight  uniformly  distributed  over  a  bar  of  cast- 
iron,  6J  feet  long,  1  inch  thick,  and  2\  deep,  will  just  be  suffi- 
cient to  break  if?  Ans.  1265.6  lbs. 

4.  A  Norway  spar,  2  inches  square,  was  broken  by  a  weight 
of  125  pounds  :  what  was  its  length  1  Ans.  94.1  in. 

PROBLEM    VI. 

22.    To  find  the  ultimate  transverse  strength  of  a  rectangular 
beam,  supported  at  both  ends,  and  loaded  at  the  middle. 

RULE. 

The  strength  is. four  times  as  great  as  that  given  by  the  last 

problem. 

Or,  r=4W»^/. 

„  '       4bd*S     ■      IW  _     ^       IW 

Hence,        1=—-^  =  —,     and     d>  = —^ 

23.  When  the  beam  is  uniformly  loaded,  the  result  will  be 
the  double  of  that  found  by  the  above  rule. 

24.  When  the  beam  is  fixed  at  both  ends,  the  result  will  be 
J  greater  than  that  found  by  the  rule. 

25.  When  the  beam  is  fixed  at  both  ends,  and  uniformly 
loaded,  the  result  will  be  double  that  in  the  last  article,  or  3 
times  that  found  by  the  above  rule. 


282  BOOK    VIII. SECTION    III. 

26.  Another  rule  is  given  by  Barlow  for  this  problem,  to 
which  the  remark  in  article  21  also  applies. 

EXAMPLES. 

1.  What  is  the  greatest  weight  that  can  be  supported  by  a 
beam  of  larch,  whose  length  is  8J  feet,  breadth  8  inches,  and 
depth  10  inches,  when  fixed  at  both  ends,  and  the  weight  uni- 
formly distributed  over  it? 

By22,  TT=4^^^4X8><1100;X1120  =  35840, 

and  by  25,  W=  3  x  35840  =  107520  lbs. 

2.  A  bar  of  cast-iron,  2  inches  square  and  15  feet  long,  is 
supported  at  both  ends  :  what  weight  applied  at  its  middle  will 
break  it?  Ans.  1440  lbs. 

3.  A  joist  of  New  England  fir  is  25  feet  long,  3  inches  thick, 
and  7  inches  deep :  what  weight  uniformly  distributed  over  it 
will  break  it  when  it  is  supported  at  both  ends? 

Ans.  4312  lbs. 

4.  The  length  of  a  plank  of  American  birch  is  10  feet,  its 
breadth  5  inches,  and  the  weight  necessary  to  break  it,  when 
supported  at  both  ends,  is  1500  lbs. :  what  is  its  thickness  ? 

Ans.  2.45  in. 

PROBLEM    VII. 

27.    To  find  the  weight  under  which  a  column  will  begin  to  bend 
when  placed  vertically  on  a  horizontal  plane. 

RULE. 

Find  the  continued  product  of  the  tabular  value  of  E;  the 
number  .2056,  the  breadth,  and  the  cube  of  the  thickness,  both 
in  inches ;  divide  this  product  by  the  square  of  the  length  in 
inches,  and  the  quotient  will  be  the  weight  in  pounds. 


STRENGTH    OF    MATERIALS.  283 

Let  b  and  t  denote  respectively  the  breadth  and  thickness, 
then  W  =. 2056  £bt3  + P. 

Hence  /*--2°56^%-       PW  and     fi-      PW 

tience,  I   _         ^       ,  b  _  ^  ^ ,      and     t  _  ^  ^. 

28.  When  the  column  is  cylindrical,  find  the  continued  pro- 
duct of  .121,  U,  and  the  fourth  power  of  the  diameter,  and 
divide  the  product  by  the  square  of  the  length. 

Let  d  =  the  diameter,  then  W  =  .121  MP+P. 

Hence,  P  = _ _,      and      *  =  -j^ 


nr  11         /   ^  10.    /10PT 

Or,        ^-rfy—,      and      ^-f^—. 

EXAMPLES. 

1.  What  weight  will  be  necessary  to  bend  a  column  of  Riga 
fir,  5  inches  square  and  8J  feet  long  ? 

TF  =  .2056  MP  *  P  =--  -2056  X  S31l4Jg°  *  5  *  5' 

=  68285  lbs. 

2.  What  weight  will  be  sufficient  to  bend  a  column  of  Mar 
forest  fir,  25  feet  long,  8  inches  thick,  and  10  broad? 

Ans.  39767  lbs. 

3.  What  weight  will  bend  a  cast-iron  column  5  inches  square 
and  6}  feet  long?  Ans.  1579000  lbs. 

4.  What  weight  can  a  column  of  pitch  pine  support,  whose 
length  is  15  feet,  breadth  12  inches,  and  thickness  10  inches? 

Ans.  380741  lbs. 

5.  What  must  be  the  length  of  a  column  of  ash,  which  is 
8  inches  thick  and  9  broad,  that  will  bend  with  a  weight  of 
822400  lbs.  ?  Ans.  81  in. 


284  BOOK    VIII. SECTION    III. 

6.  Under  what  weight  will  a  cylindrical  column  of  Riga  fir 
begin  to  bend,  its  diameter  being  10  inches,  and  length  16^ 
feet?  Ans.  160748  lbs. 

7.  A  cylindrical  pillar  of  cast-iron,  7^-  inches  in  diameter, 
begins  to  bend  under  a  weight  of  160000  lbs. :  required  its 
length.  Ans.  34i/£. 

29.  Another  solution  of  this  problem  can  be  obtained  by 
employing  the  modulus  of  elasticity  for  the  given  material.  For 
a  rectangular  beam  we  have  this 

RULE. 

Find  the  continued  product  of  .8225,  the  tabular  value  of 
M,  and  the  square  of  the  thickness  in  inches,  and  divide  this 
product  by  the  square  of  the  length  also  in  inches ;  then  the 
quotient  will  be  the  length  of  a  bar  of  the  same  thickness, 
breadth,  and  material,  whose  weight  will  bend  the  given  column. 

Let  b  =  the  thickness,  and  M  ==  the  modulus  of  elasticity, 
then  if  L  =  the  required  length  of  the  bar, 

*Z  =  .8225  J/62  -i-  P  =  .8225  M  (jj 

For  the  example  given  above, 

The  solidity  of  this  length  of  the  column  is 

v  —  bdl  —  T52  X  T52  X  8288.5  =  1439  cubic  feet. 

The  specific  gravity  of  which  is  750 ;  and  hence  (9)  its 
weight  W=  750  v  =  750  X  1439  =  1079250  ounces  =  67453  lbs. 

The  weight  found  by  the  preceding  rule  was  68285,  and  the 
difference  between  these  results  is  832,  or  less  than  g^  part  of 
the  whole. 

*  For  the  theoretical  investigation  of  this  rule,  see  Whewell's  Dynamic^ 
art.  147. 


„  .2      .8225  X  4080000  x  52 

.8225  M[  —  \= — =  8288.5  feet. 

100^ 


STRENGTH    OF    MATERIALS.  285 

The  modulus  Jtfmay  be  found  in  various  ways ,  but  after  E 
is  determined,  the  value  of  M  can  be  found  by  multiplying  E 
by  576,  and  dividing  the  product  by  the  specific  gravity  of  the 

E 
material;  that  is,  M ■=.  576—.    The  results  obtained  by  differ- 
ent nfethods  will,  however,  differ  by  a  small  fraction  of  the 
whole,  in  consequence  of  the  variations  in  the  qualities  of  the 
same  kind  of  materials. 

The  reader  may  solve  the  exercises  of  the  preceding  prob- 
lem by  this  method,  and  compare  the  results. 

30.  In  applying  the  preceding  rules  to  cast-iron,  the  results 
are  considerably  different  from  those  obtained  by  the  rules 
given  by  Tredgold  in  his  essay  on  the  strength  of  cast-iron. 
The  cause  of  this  appears  to  be  that  the  absolute  strength  of 
the  metal  determined  by  him  is  considerably  lower  than  the 
value  adopted  by  some  others ;  for,  as  remarked  by  Dr.  Young, 
"  a  permanent  alteration  of  form  limits  the  strength  of  mate- 
rials almost  as  much  as  fracture,  since  in  general  the  force 
which  is  capable  of  producing  this  effect,  is  sufficient,  with  a 
small  addition,  to  increase  it  till  fracture  takes  place ;"  and 
Tredgold  adopted  as  the  measure  of  the  .strength  a  force  less 
than  that  capable  of  producing  permanent  alteration  of  form. 
He  found,  by  a  comparison  of  many  experiments,  that  the  force 
necessary  to  break  a  beam  of  cast-iron  is  3.3  times  greater 
than  that  which  is  sufficient  to  produce  this  permanent  change. 
The  cast-iron  to  which  the  rules  given  by  him  apply,  is  soft 
grey  cast-iron,  which  is  possessed  of  considerable  malleability, 
a  criterion  of  its  superior  strength  in  resisting  fracture  by  a 
shock  or  blow,  to  which  the  white  hard  iron  is  liable  on  ac- 
count of  its  brittleness,  and  the  former  is  not  liable  to  sudden 
failure.     This  grey  iron  was  found  to  be  capable  of  supporting 


286  BOOK    VIII. SECTION    III. 

15300  lbs.  of  a  longitudinal  strain  for  every  square  inch  of 
section,  without  being  subject  to  permanent  change  of  form. 
Most  of  the  following  rules  are  deduced  from  those  given  by 
Tredgold  in  his  essay. 

PROBLEM    VIII. 

31.    To  find  the  weight  that  a  rectangular  cast-iron  beam,  sup- 
ported at  both  ends,  can  sustain  at  its  middle. 

RULE. 

Find  the  continued  product  of  850,  the  breadth  and  square 
of  the  depth,  both  in  inches,  and  divide  this  product  by  the 
length  in  feet,  and  the  quotient  will  be  the  required  weight. 

That  is,  W=  850  bd*~l. 

„  _      850  6^  IW  .      __        IW 

Hence,     /  =  __,6  =  _,      and     e*>  =  — . 

32.  For  malleable  iron,  use  950  instead  of  850.  For  a  beam 
fixed  at  both  ends,  apply  the  rules  in  arts.  24,  25.  The  weight 
of  the  beam  must  always  be  added  to  the  applied  weight :  the 
weight  of  the  beam  is  equivalent  to  f-  of  it  applied  at  the 
middle ;  and  any  weight  uniformly  distributed  is  also  equiva 
lent  to  |-  of  it  applied  at  the  middle. 

EXAMPLES. 

1.  A  bar  of  cast-iron  is  2  inches  square  and  15  feet  long: 
what  weight  will  it  be  capable  of  supporting  1 

vr      QKuLn '■'.  7      850  x  %X  22      170x8 

W  =  850  bd2  -'-  I  = — = —  =  453  lbs. 

15  3 

This  result  is  only  about  J  of  that  found  by  the  former 
method  in  art.  22  (see  the  second  example),  for  the  reason 
explained  in  art.  30. 

2.  Find  the  weight  that  can  be  supported  by  a  beam  5  inches 
square  and  10  feet  long?  *        Ans.  10625. 


STRENGTH    OF    MATERIALS.  287 

3.  A  beam  of  cast-iron  is  20  feet  long  and  2£  inches  broad, 
and  it  has  to  support  a  load  of  10000  lbs. :  what  must  be  its 
depth?  Ans.  9  J. 

4.  A  cast-iron  joist  is  30  feet  long,  10  inches  deep,  and  3 
inches  broad  :  what  weight,  uniformly  distributed,  can  it  sus- 
tain? Ans.  13600  lbs. 

PROBLEM    IX. 

To  find  the  weight  that  a  beam  fixed  at  one  end  can  sustain 
at  its  free  end. 

RULE. 

The  weight  is  J  of  that  found  by  the  preceding  problem. 
W=i  X  850  6rf2-rZ  =  2126^  +  1. 

The  other  formulas  are  the  same  as  in  the  last  problem,  if 
212  is  used  instead  of  850  ;  and  for  wrought  iron,  use  \  x  950, 
or  238  instead  of  212. 

34.  When  the  weight  is  uniformly  distributed  over  the  beam, 
take  \  of  that  found  by  the  preceding  problem.  Or  in  the  for- 
mulas of  last  problem  use  425  for  850. 

EXAMPLES. 

1.  A  beam  is  30  feet  long,  8  inches  deep,  and  2£  broad: 
what  weight  can  it  support?  Ans.  1132  lbs. 

2.  What  load  uniformly  distributed  over  a  ^eam  32  feet 
long,  4  inches  deep,  and  2  broad,  can  it  sustain 

Ans.  425  lbs. 

3.  A  beam  20  feet  long  and  10  inches  deep  supports  a  load 
of  17000  lbs. :  what  is  its  breadth?   '  Ans.  16  in. 

4.  A  beam  24  feet  long  and  2  inches  broad  supports  1735 
lbs.  uniformly  distributed  :  required  its  depth. 

Ans.  7  in. 


288  ROOK    VIII. SECTION    III. 


PROBLEM    X. 


35.    To  find  the  weight  which  a  solid  cylinder  of  cast  iron,  sup- 
ported at  both  ends,  can  sustain  at  the  middle. 

RULE. 

Multiply  the  cube  of  the  diameter  in  inches  by  500,  and 

divide  the  product  by  the  length  in  feet,  then  the  quotient  will 

be  the  weight. 

W  =  500  d3  ~  I,  where  d  =  the  diameter. 

,      500^  ,      „       IW 

Hence,  1  =  -^-,      and     ^  = —. 

36.  When  the  weight  is  uniformly  diffused,  it  will  be  double 
that  given  by  the  preceding  rule;  or, 

Txr      1000 <P  lOOOcF  i 

W  =z - —  ;  hence  I  =  — — - ,      and     d  =  — -  ylW. 

I  W  10 

Since  |4p£  =  1$  =  \  nearly,  the  weight  sustained  by  a  cylin- 
der is  nearly  equal  to  §  of  that  sustained  by  a  square  beam  of 
the  same  length,  and  whose  depth  is  equal  to  the  diameter  of 
the  cylinder. 

EXAMPLES. 

1.  What  weight  will  a  cylinder  10  feet  long  and  4  inches 
diameter  support  1  Ans.  3200  lbs. 

2.  What  weight  will  a  uniformly  loaded  cylinder  support,  its 
length  being  24  feet,  and  diameter  10  inches  ? 

Ans.  41667. 

3.  What  will  be  the  diameter  of  a  cylinder  20  feet  long, 
which  is  capable  of  supporting  3134  lbs.  ?  Ans.  5  i», 

4.  What  will  be  the  limit  of  the  length  of  a  cylinder  uni- 
formly loaded  by  a  weight  of  100000  lbs.,  whose  diameter  is 
12  inches?  *  A ns.  17.28//. 


STRENGTH    OF    MATERIALS.  289 

PROBLEM    XI. 

37.  To  find  the  weight  that  a  cylinder,  fixed  at  one  end,  can 

sustain  at  the  free  end. 

RULE. 

Multiply  the  cube  of  the  diameter  in  inches  by  125,  and 
divide  the  product  by  the  length  in  feet,  and  the  quotient  will 
be  the  weight. 

.    W=\2hd?  +  l,     or      W=  A  X  500^  ~  I. 

The  weight  is  just  the  fourth  of  that  found  in  art.  35,  and  the 

formulas  the  same  as  in  that  article,  if  125  =  -J- of  500  is  taken 

for  500. 

,       125  d?  ■  ■,    -       IW  1  ,__ 

^=~F"J    and    ^  =  T25'     or    d  =  ~EV^ 

EXAMPLES. 

1.  What  weight  will  a  cylinder  10  feet  long  and  4  inches 
diameter  support  at  its  free  end?  Ans.  800  lbs. 

2.  What  will  be  the  diameter  of  a  cylinder  20  feet  long  that 
can  support  784  lbs.  %  Ans.  5  in. 

3.  What  will  be  the  length  of  a  cylinder,  which  is  12  inches 
diameter,  that  supports  12500  lbs.  ?  Ans.  17.28  ft. 

PROBLEM    XII. 

38.  To  find  the  exterior  diameter  of  a  holloio  cylinder  of  cast- 
iron,  supported  at  both  ends,  to  sustain  a  weight  applied  at  the 
middle,  the  ratio  of  the  interior  and  exterior  diameters  being 
given. 

RULE. 

Let  the  ratio  of  the  exterior  to  the  interior  diameter  be  that 
of  1  to  n ;  then  take  the  difference  between  1  and  the  fourth 
power  of  n,  and  multiply  it  by  500;  find  also  the  product  of 

13 


290  BOOK    VIII. SECTION    III. 

the  length  and  the  weight;  divide  the  latter  product  by  the 
former;  then  the  quotient  will  be  the  cube  of  the  diameter. 
d3  =  lW~-  500(1  -n*). 
When  the  exterior  diameter  d  is  found,  the  interior  diameter 
will  be  obtained  by  multiplying  d  by  n.     If  d'  =  the  interior 
diametei,  and  t  =  the  thickness,  of  the  metal,  then 

d'  =  nd,  and  t  =  $  (d  -  d')  =  \  (1  -  n)  d. 

EXAMPLES. 

1.  The  weight  supported  by  a  hollow  cylinder  is  32000  lbs., 
its  length  is  12  feet,  and  the  ratio  of  the  exterior  and  interior 
diameters  is  10  to  1  :  what  are  its  diameters? 

<P  -  IW^  500  U  -  »«,  -   12  x  3200°  -   12xC4 
d-lW.M0(l      »)-50o(1_J4)-1_.00oT 

=  -q^  =  -^ ■  =  868'9'     aDd     d  =  V^&V  =  9.5  inches. 
.yyy         y«j«/ 

Hence,  &'  =  nd  t=  .1  x  9.5  =  .95  inch. 

and    t  =  ^  (d  -  d')  =  \  (9.5  -  .95)  =  \  X  8.55  =  4.27; 

or  t  =  J  (1  -  n)  rf  =  £  (1  -.1)  X  9.5  =  \  X.9  X  9.5  =  4.27. 

2.  A  hollow  cylinder  10  feet  long  supports  2500  lbs.,  the 
ratio  of  its  diameters  is  2  to  1  :  what  are  the  diameters  % 

Ans.  1.88  and  3.76,  and  thickness  of  metal  .94  in. 
39.  If  the  thickness  of  the  metal  is  to  be  -J  of  the  exterior 
diameter,  then 

For  the  equation  t  =  -J  (1  —  n)  d  becomes  \d  =  J  (1  —  n)  d; 
hence,  2  =  5  —  5n,  and  »  =  f  =  .6,  and  1  —  w4=  1  —.1290 
=  .8704,  500  (1  -  n*)  z=  500  x  .8704  =  435,  and  ^Q7  =  .132. 

3.  A  hollow  cylinder  9  feet  long  is  intended  to  support 
15000  lbs.,  and  the  thickness  of  the  metal  is  to  be  £  of  the 
exterior  diameter  :  reauired  its  diameters. 

Ans.  C>.8  and  4.1  in. 


STRENGTH    OF    MATERIALS  291 

PROBLEM   XIII. 

4C.   To  find  the  deflexion  of  a  rectangular  beam  of  cast-iron, 
supported  at  the  ends,  aud  fully  loaded  in  the  middle. 

RULE. 

Divide  the  square  of  the  length  in  feet  by  50  times  the 
depth  in  inches,  and  the  quotient  is  the  deflexion  in  inches. 
Let  D  =  the  deflexion, 

then  J)  =  ~^-1,P  =  50Dd,     and     d  =  -£-=. 

50  a  D\)  1/ 

The  cohesive  strength  of  the  beam  is  considered  to  be  equi- 
valent to  15300  lbs.  on  the  square  inch. 

EXAMPLES. 

1.  Find  the  deflexion  of  a  beam  18  feet  long  and  12  inches 
deep.  Ans.  .54  in. 

2.  What  is  the  deflexion  of  a  beam  30  feet  long  and  15 
inches  deep?  Ans.  1.2  in. 

PROBLEM   xiv. 

41.     To  find  the  deflexion  of  a  rectangular  learn  of  cast-iron, 
supported  at  both  ends,  when  fully  loaded,  and  the  weight  uni- 
formly distributed  over  it. 

RULE. 

Divide  the  square  of  the  length  in  feet  by  40  times  the 
depth  in  inches,  and  the  quotient  is  the  deflexion  in  inches. 

D  =  ^-,  I2  =  40  JJd,     and     d  = 


40  <?  '  ~40D' 

EXAMPLES. 

1.  What  is  the  deflexion  of  a  beam  12  feet  long  and  5  inches 
deep?  Ans.  .72  in. 

2.  Find  the  deflexion  of  a  beam  15  feet  long  and  5J-  inches 
deep.  Aiis.  1.07  in. 


292  BOOK    VIII. SECTION    III. 

PROBLEM    XV. 

42.  Tojind  the  resistance  to  torsion  of  a  square  shaft  of  cast  iron* 

RULE. 

Find  the  continued  product  of  92.5,  the  number  of  degrees 
in  the  angle  of  flexure,  and  the  fourth  power  of  the  side  in 
inches;  divide  this  product  by  the  product  of  the  length  and 
the  leverage  of  the  power  both  in  feet,  and  the  quotient  will  be 
the  resistance  in  pounds. 

Let  a  z=  the  number  of  degrees  in  the  angle  of  flexure, 
and       s  =  the  side  of  the  shaft  in  inches, 
I  =  the  length  in  feet, 
L  =  the  leverage  of  the  power  W  in  feet, 
W=  the  resistance  in  pounds  ;\ 
then    W  =92.5  as*  +  IL. 
Hence, 

LWl      _  92.5  as*       _  92.5  as*  LWl 

a-W^s~*'l-~LW''L-~TW~''     And     S   "92.5a 

EXAMPLES. 

1.  The  side  of  a  square  shaft  is  3  inches  and  its  length  12 
feet,  and  it  is  driven  by  a  power  acting  with  a  leverage  of  2 
feet :  what  power  may  be  applied  to  it,  so  as  not  to  cause  a 
flexure  of  more  than  1°  C?  Ans.  343.4  lbs. 

2.  What  must  be  the  side  of  a  square  shaft  of  cast-iron  10 
feet  long  to  resist  a  power  of  1500  lbs.  acting  with  a  leverage 
of  3  feet,  with  an  angle  of  flexure  of  1£  degrees'? 

Ans.  4.244. 

PROBLEM    XVI. 

43.  To  find  the  resistance  to  torsion  of  a  solid  cylinder  of  cast-iron, 

RULE. 

Find  the  continued  product  of  55,  the  number  of  degree?  in 
the  angle  of  flexure,  and  the  fourth  power  of  the  diameter  in 


STRENGTH    OF    MATERIALS.  293 

inches ;  divide  this  product  by  the  product  of  the  length  and 
the  leverage  both  in  feet,  and  the  quotient  will  be  the  resis- 
tance. 

W  =  55ad*  +  IL. 

LWl  .      55 ad*    T       bbad*  J±      LWl 

Ben<»,a  =  mL,l  =  TW-9L  =  -iW- ,      and      *  =  -^ 

.55  11        22        2  ... 

Since  — -  ==  —-=  =  —  =  —  nearly,  a  cylinder  can  sustain 

«7/*.0         lo.O        o/  o 

nearly  §■  of  the  torsion  that  a  square  beam  is  capable  of,  whose 
side  is  equal  to  the  diameter  of  the  cylinder. 

EXAMPLES. 

1.  A  shaft  is  35  feet  long  and  10  inches  diameter:  what 
power  can  it  sustain,  acting  at  a  leverage  of  2  feet  with  an 
angle  of  flexure  of  1°  ?  Ans.  7857  lbs. 

2.  A  shaft  is  20  feet  long,  and  has  to  transmit  a  power  of 
4500  lbs.,  acting  with  a  leverage  of  2  feet :  what  must  be  its 
diameter  so  that  the  angle  of  flexure  may  be  1^°  1 

Ans.  6.8  in. 
44.  The  resistance  to  a  crushing  force  cannot  be  determined 
with  even  an  approximation  to  the  truth,  when  the  length  is 
small.  The  theory  of  the  subject  is  in  a  very  imperfect  state ; 
and  the  experiments  regarding  it  are  equally  deficient,  either  in 
accuracy  or  on  account  of  the  smallness  of  the  specimens.  The 
following  are  some  of  the  results  obtained  by  Rennie,  showing 
the  force  necessary  to  crush  different  kinds  of  materials  :* 

A  cube  of  cast-iron  \  inch,  ....     10561  lbs. 

A  cube  of  cast-iron  ■£  inch,        ...        -  1454  . . 

A  piece  £  inch  square  and  -J  inch  long,  -        -       9314  . . 

A  piece  £  inch  square  and  1  inch  long,      -         •  6321  . . 

*  Tredgold's  Essay  on  the  Strength  of  Cast  Iron,  art  64;  and  Leslie  a 
Natural  Philosophy,  p.  221. 


294 


BOOK    VIII. SECTION    HI, 

A  cube  of  elm  of  1  inch, 1284  . . 

A  cube  of  deal  of  1  inch,          -        -        -        -  1928  . . 

A  cube  of  English  oak  1  inch,  -         -         -  8800  . . 

A  cube  of  Craigleith  stone  of  1-J-  inch,  crushed  in 

direction  of  strata, 15560  .  , 

The  same  crushed  across  the  strata,  -        -  12346  . . 

Cornish  granite,  of  same  size,       -  -  14302  . . 

Peterhead  granite,        do.         -        -  -  18636  . . 

Aberdeen  blue  granite,  do.  ....  24536. 


PROBLEM    XVII. 

45.   To  find  the  weight  that  could  be  safely  supported  by  a  column 
of  cast-iron  of  considerable  length  resting  on  a  horizontal  plane. 

RULE. 

The  rules  for  this  problem  would  be  very  tedious,  if  ex- 
pressed in  common  language ;  they  are,  therefore,  given  alge- 
braically.* 

Let  W  =  the  weight  in  pounds,  b  =  the  greater  side,  and 
d  =  the  less  side  both  in  inches,  and  I  =  the  length  in  feet ; 

15300  bd* 


then 


W. 


dz  XASI2 
when  the  weight  acts  along  the  axis. 

When  the  weight  acts  at  the  distance  a  inches  from  the  axis, 
15300  fcd3 


then 


W 


d2  +  6  ad  +.18  I2 
For  a  cylinder,  the  force  acting  in  the  direction  of  its  axis, 

9562  d* 
d?  +  .18  I2' 
When  the  direction  of  the  force  is  a  inches  from  the  axis, 

9562  d4 


W=  , 


d2  +  Q>ad  +  ASl2 


See  Tredgold's  Essay,  articles  240—247. 


STRENGTH    OF    MATERIALS.  295 

It  is  consideied  to  be  probable  that  the  weight  generally  acts 
at  the  edge  which  is  nearest  to  the  axis,  that  is,  at  the  distance 
\d\  hence,  a  =  |t/,  and  then 

lor  a  rectangular  column  or  cast-iron.  W  = 


for  a  rectangular  column  of  malleable  iron,  W  = 

for  a  rectangular  column  of  oak.  W  = 

for  a  cylinder  of  cast-iron,  W  =. 

for  a  cylinder  of  malleable  iron,  W  ■ 

for  a  cylinder  of  oak,  W : 


4tf2  +  .18Z2 

17800  bd* 
Td?  +  .16/2 

3960  bd3 
4  d2  +  .5  J2 

9562  rf* 
4:d*  +  T8P 

11125  # 
4  rf2  -f  7l6T* 

2470  d* 
4d2  +  .5P 


46.  As  the  pressure  has  the  greatest  effect  in  bending  a  pil- 
lar when  it  acts  farthest  distant  from  the  centre,  that  is,  at  the 
edge,  it  is  safest  in  practice  to  compute  the  strength  by  the  last 
six  formulas. 

In  practice,  for  safety,  a  beam  ought  not  to  be  subjected  to  a 
pressure  exceeding  J-  or  J  of  its  strength,  as  determined  by 
computation. 

EXAMPLES. 

1.  Find  the  weight  that  can  be  safely  supported  by  a  square 
pillar  5  inches  in  the  side  and  6J  feet  long,  the  direction  of  the 
force  being  alcng  the  axis.  Ans.  298537. 

2.  What  weight  can  a  rectangular  column  of  malleable  iron 
6  inches  broad,  5  thick,  and  12J  feet  long,  support,  supposing 
the  pressure  to  be  at  the  edge  nearest  the  axis  ? 

Ans.  106800  U>8. 

3.  What  weight  can  a  cylindrical  pillar  of  oak  10  feet  1  ng 


'29G  BOOK    VIII. SECTION    III. 

and  10  inches  in  diameter  support,  the  pressure  being  at  the 
edge  of  the  top  1  Ans.  54889. 

4.  Find  the  pressure  that  can  be  sustained  by  the  piston-rod 
of  a  steam-engine,  its  length  being  4  feet,  and  its  diameter  4 
inches.  Ans.  42789  lbs. 

47.  In  experimenting  on  transverse  strains  of  materials,  it  is 
found,  as  formerly  stated,  that  a  certain  amount  of  straining 
force  produces  a  set,  or  permanent  change  of  form..  This  state 
must  result  from  an  entire  destruction  of  the  elasticity  by  the 
overstraining  or  setting  force ;  and,  as  it  cannot  take  place 
suddenly,  it  follows  that  the  elasticity  must  gradually  diminish 
as  the  force  increases,  till  the  material  attains  this  state.  •  From 
what  point  this  diminution  of  elasticity  begins,  is  uncertain ; 
but  it  is  very  probable  that  it  begins  with  the  effect  of  the 
smallest  straining  force,  though  at  first  the  diminution  may  be 
insensible,  as  the  distention  itself  is  very  small ;  so  that  the 
elasticity  may  go  on  gradually,  perhaps  nearly  uniformly, 
diminishing  till  the  force  becomes  a  setting  force,  and  com- 
pletely destroys  it. 

If  this  is  the  law  of  diminution  of  elasticity,  then  the  forces 
producing  fracture  cannot  be  considered  as  having  any  constant 
proportion  to  what  may  be  called  the  practical  strength  of  the 
material,  as  measured  by  the  greatest  force  that  is  just  insuffi- 
cient to  overstrain  it ;  for  one  of  the  elements  concerned  in  the 
estimate  of  the  practical  strength,  namely,  the  elasticity,  ceases 
to  exist  even  when  the  force  is  less  than  that  required  to  pro- 
duce fracture.  This  view  of  the  subject  will  assist  in  the  ex- 
planation of  the  rather  anomalous  results  obtained  by  different 
experimentalists. 


A  TABLE 

OF 

LOGARITHMS  OF  NUMBERS 

FROM   1  TO   10,000. 


N. 

■    Log. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

, 

0- 000000 

26 

1 -4I4Q73 

5i 

1-707570 

76 

1-880814 

2 

o«3oio3o 

27 

i-43i364 

52 

1-716003 

]l 

1-886491 

3 

o-477'2i 

28 

1 -447i58 

53 

1-724276 

1 -892093 

4 

0-602060 

29 

1-462398 

54 

1 -732394 

S 

1 -897627 

5 

0-698970 

3o 

1-477121 

55 

i-74o363 

1 .903090 

6 

0-778151 
0-845098 

3i 

1 -491362 

56 

1-748188 

81 

I-Oo84b5 

I 

32 

i-5o5i5o 

57 

1-755875 

82 

i-9i38i4 

0-903090 

33 

i-5i85i4 

58 

1-763428 

83 

1-919078 

9 

0-954243 

34 

1 -53 1479 

59 

1 -770852 

84 

1-924279 

10 

1 • 000000 

35 

1 • 544o68 

60 

i-778i5i 

85 

1-929419 
1-934498 

1 1 

1-041393 

36 

i-5563o3 

61 

i-78533o 

86 

12 

1 -079181 
i • 1 1 3943 

8 

1-568202 

62 

1 -792392 

87 

1 -939519 
1 • 944483 

i3 

1-579784 

63  . 

1-799341 

88 

14 

1-146128 

39 

1 -591065 

64 

1-806181 

89 

1 -949390 

i5 

1 -176091 

40 

1 -602060 

65 

i-8i2oi3 
1-819544 

90 

1-954243 

16 

1 • 2041 20 

41 

1-612784 

66 

91 

1 -959041 
1-963788 

n 

1 • 230449 
1 -255273 

42 

1 -623249 

67 

1-826075 

92 

18 

43 

1-633468 

68 

i-8325o9 

93 

1-968483 

J9 

1-278754 

44 

1-643453 

69 

1-838849 
i- 845098 
1 -85i258 

94 

1 -973128 

20 

i-3oio3o 

45 

i-6532i3 

70 

95 

1-977724 

21 

1.322219 

46 

i-662758 

7i 

96 

1-982271 

22 

1-342423 

% 

1 -672098 

72 

1-857333 

u 

1-986772 

23 

1. 361728 

1 -681241 

73 

1-863323 

1-991226 

24 

i-38o2ii 

49 

1 -690196 

74 

1 -86o23 2 
1 -875061 

99 

1-995635 

75 

1 • 397940 

5o 

1-698970 

75 

100 

2 • 000000 

Kemark.  In  the  following  table,  in  the  nine  right 
hand  columns  of  each  page,  where  the  first  or  lead- 
ing figures  change  from  9's  to  O's,  points  or  dots  are 
introduced  instead  of  the  O's,  to  catch  the  eye,  and  to 
indicate  that  from  thence  the  two  figures  of  the  Log- 
arithm to  be  taken  from  the  second  column,  stand  in 
the  next  line  below. 


2 

A  TABLE 

OF  LOGARITHMS  FROM  1 

TO 

10,00Q. 

N. 

0 

1 

2 

3  |  4  |  5  |  6  |  7  |  8 

9 

D. 

100 

000000 

0434 

^0868 

i3oij  1734  2166;  2598!  3029I  3461 

389i 

432 

101 

4321 

475i 

5i8i 

5609:  6o33  6466;  6894  732 1  j  7748 

8174 

428 

102 

8600 

9026 
325g 

Q45 1 

368o 

9876 

®3oo 

•724!  1 147;  1570  1993 

24i5 

424 

io3 

012837 

4100 

452i 

4940  536o!  5779 

6197 

6616 

419 
416 

1 04 

7o33 

745 1  7 

8284 

8700 

91 16  o532  9947 
3252  3664:  4075 

•36 1 

•775 
.4896 

io5 

021 189 
53o6 

i6o3 

2016 

2428 

2841 

4486 

412 

106 

5715 

6i25 

6533 

6942 

735o 

7757  8164 

857i 

8978 

408 

108 

o384 
o33424 

9789 
3826 

•195 

•600 

1004 

1408 

1812  2216 

2619 

3021 

404 

4227 
8223 

4628 

5029 

543o 

583o  623o 

6629 

7028 

400 

109 

742£ 

7825 

8620 

9017 

94i4 
3362 

'  981 1  j  »207 
3755  4U8 

•602 

•908 

396 

no 

041393 

1787 

2182 

2576 

6885 

454o 

4932 
883o 

393 
389 

m 

5323 

57i4 

6io5 

6495 

7275 

7664!  8o53 

8442 

112 

9218 
05J078 

9606 
3463 

9993 

•38o 

•766 

ei53 

i538;  1924 

2309 

2694 

386 

n3 

3846 

423o 

46i3 

4996  5378  5760 
88o5  9i85  o563 
2582  2Q58!  3333 

6142 

6524 

382 

114 

6905 

7286 

7666 

8046 

8426 

9942 
3709 

•320 

379 
376 

n5 

060698 
4458 

1075 

U52 

1829 

2206 

4o83 

116 

4832 

52o6 

558o 

5953 

6326 

6699 1  7071 

7443 

78i5 

372 

117 

8186 

8557 

8928 

9208 
2985 

0668 
3352 

••38 

•407  '776 

1 U5 

i5i4 

369 

118 

071882 

225o 

2617 

37i8 

4o85  445i 

4816 

5i82 

366 

119 

5547 

5oi2 
oD43 
3i44 

6276 

6640 

7004 

7368 

77311  8094 
i347  1707 

8457 

8819 

363 

120 

079181 

9904 
35o3 

•266 

•626 

•987 

2067 

2426 

■  36o 

121 

082785 

386 1 

4219 

4576  4934:  5291 

5647 
9198 

6004 

357 

122 

636o 

6716 

7071 

7426 

7781 

8 1 36  8490]  8845 

9552 

355 

123 

9905 
093422 

•258 

•611 

•963 

i3i5 

1667 

20181  2370 

2721 

3071 

35i 

124 

3772 

7257 

4122 

447i 

4820 

5i69 

55 1 8  5866 

62i5 

6562 

349 

125 

6910 
100371 

7604 

795i 

8298 

8644 

8990  9335 

0681 
3 1 19 

••26 

346 

126 

0715 

1059 

i4o3 

1747 

2091  2434  2777 

3462 

343 

IS 

38o4 

4U6 

4487 

4828 

5169 
8565 

55io  585i  6191 

653 1 

6871 

34o 

7210 

7549 

7888 

8227 

89o3 1  9241  9579 

9916 
3275 

•253 

338 

129 

1 1 0590 

0926 

1263 

1 599 

1934 

2270I  26o5j  2940 

3609 

335 

i3o 

1 1 3943 

4277 

461 1 

4944 

5278 

56n 

5g43  6276 

6608 

6940 

333 

i3i 

7271 

76o3 

7934 

8265 

8595 

8926 

9256 

9586 

o9[5 

•245 

33o 

132 

120574 

0903 

1 23  I 

i56o 

1888 

2216 

2544 

2871 

3i98 
6456 

3525 

328 

1 33 

3852 

4178 

45o4 

483o 

5i56 

548i 

58o6 

6i3i 

6781 

325 

i34 

7io5 

7429 
o655 

7753 

8076 

8399 

8722  9045  9368 

9690 

••12 

323 

i35 

i3o334 

0977 

1298 

1619 

1939'  2260  258o 

2900 

3219 

321 

1 36 

3539 

3858 

4.77 

4496 

4814 

5 1 3 3  545i  5769 

6086 

64o3 

3i8 

l3Z 
i38 

6721 

7037 

7354 

7671 

7987 

83o3|  86181  8934 

9249 

9564 

3i5 

1 43oiD 

•i94 

•5o8 

•822 

n36 

i45o  1763!  2076 
4574I  4885}  5i96 

2389 

2702 

3 1 4 

139 

3327 

0438 

363o 

6748 

3o5i 

4263 

55o7 

58i8 

3n 

140 

146128 

7o58 

7367 

7676J  7985  8294 

86o3 

89.1 

309 

141 

9219 
152288 

9527 

9835 

•142 

•449 

•756  io63|  1370 

1676 

1982 

307 

142 

2594 

2900 

32o5 

35io 

38i5  4120;  4424 

4728 

5o32 

3o5 

143 

5336 

564o  5943 

6246 

6349 

6852  7 i 54  7457 

7759 

8061 

3o3 

144 

8362 

8664  8965 

9266 

9567 

9868!  '^S!  '469 

•769 
-^58 

1068 

3oi 

145 

161 368 

1667;  1967 

2266 

2564 

2863  3i6i!  3460 

4o55 

299 

146 

4353 

465o  4947 

5244 

5541 

5838  61 34  643o 

6726 

7022 

297 
295 

148 

7317 

76i3  7908 

8203 

8497 

8792 1  9086  9380 

9674,  9968 

170262 

o555  0848 

1141 

1434 

1726;  2019  23ll 

26o3 

2895 

293 

149 

3i86 

3478 

3769 

4060 

435 1 

4641 1  4g32  5222 

55i2 

58o2 

291 
289 

i5o 

1 7609 1 

638i 

6670 

6o59 
9839 

7248 

7536 

7825:  8n3 

8401 

8689 
1 558 

i5i 

8077 
181844 

9264 

9552 

•126 

•4i3 

•699'  «985 
3555  3839 

1272 

287 

285 

l52 

2129 

24i5 

2760 

2985 
5825 

3270 

4i23 

4407 

1 53 

4691 

4975  5s59 

78o3 j  8084 

55.',2 

6108 

63gi  6674 

6956 

7239 

283 

1 54 

7521 

8366 

8647 

8928 

9209  9490 
2010  2289 

9771 

••5 1 

281 

1 55 

190332 

06 1 2  0892 

1171 

I45i 

n3o 

2  567 

2846 

279 

1 56 

3i25  34o3  368i 

3939  4237 

45i4|  4-92;  5069 

5346  5623 

278 

1 57 

5899 ]   6176  6453 
8657!  8932 !  9206 

6729  too5 

7281 1  7556  7832  810-  8382 

2^6 

1 58 

948'  9755  ••29I  «3o3  »577  *85o 

1124 

274 

l59 

201397 

1670J  1943 

2216  2488,  2761  j  3o33|  33o5j  3577 

384c 

272 

N. 

0 

.1.1 

3  I  4  "  |  5  |  6  |  7  1  8 

9 

D. 

A  TABLE 

OF  ] 

LOGARITHMS  FROM  1 

TO 

10,000. 

n 

N.  | 

0    |    ,  |  , 

3  1 

* 

£j  ■ 6  |  7 

8  1 

9 

D. 

1 60 

204120;  4391 

4663 

4934 

D204 

5475  5746  6016 

6286; 

6556  271 

l6l 

6826'  7006 

7365 

7^34! 

7904 

8173  8441 

8710 

80-9 

9247'  2*69 

l62 

90 1 5  9783  e#5i 

•319 

•586 

•853!  1121 

i388 

1 654 

192 1  267 
4379  266 

1 63 

212188  2454  2720 

29% 

3252 

35i8  3783  4049 

43i4 

164 

4844  5io9  5373 

5638 

5902 

6166  643  0  6694 

6907 

7221  264 

i65 

7484  7747  8010 

8273; 

8536 

8798  9060  9323  9585! 

9846  262 

166 

220108  0370  o63i 

0892 

1 1 53 

1414  1675,  io36 
4oi5  4274  4533 

2196 

2456  261 

167 

2716  2976  3236 

3496 
6084 ! 

3755 

4792i 

5o5i  259 
763o  258 

168 

53o9:  5568 

5826 

6342 

6600  6858(  71 1 5 

7372 

169 

7887;  8144 

8400 

8657j 

8gi3 

9170.  9426,  9682 

9938 

•193]  256 

170 

230449  0704 

0960 

I2l5! 

1470 

1724!  1979  2234 

2488 

2742  254 

■71 

2996 

5528 

325o 

3do4 

3757| 

401 1 

4264  4017 

4770   D023; 

5276I  253 

172 

578i 

6o33 

6285, 

6537 

6789;  7041 

7292  7544' 

7795;  252 

i73 

8046 

8297 

8548 

8799 

9049 

9299  955o 

9800 

••5o; 

•3oo|  25o 

174 

240549  0799 
3o38J  3286 

1048 

1297; 

3782; 

1 546 

1793  2044 

2293 

2541 

2790  249 
5266  248 

i75 

3534 

4o3o 

4277!  4525 
6745;  6991 

4772 

5019 

176 

55i3|  5759 

6006 

6252 

6499 
8954 

7237 

7482; 

7728,  246 

H7 

7973  8219 

8464 

87o9; 

9198  9443 
i638;  1881 

9687 

9932 

•1761  245 

178 

25o420j  0664 

0908 
3338 

ii5i 

i395 

3S22 

2125 

2368, 

2610  243 

H9 

2853  3o96 

358o 

4o64!  43o6 

4548  479°; 

5o3i  242 

180 

255273|  55i4 

5755 

5996! 

6237 

6477  6718 

6958  71981 

7439  241 

181 

7679!  79i8 

8,58 

83g8 

8637 

8877;  9"6 

9355 

9594 

9833  239 
2214  238 

182 

260071  o3io 

o548 

0787: 

1025 

1263  i5oi 

1739 

1976; 

1 83 

245i  2688 

2925 

3162 

3399 

3636|  3873 

4109 

4346: 

4582  237 

184 

4818  5o54 

5290 

5525j 

576i 

5996!  6232 

6467 

6702 

6937  235 

1 85 

7172 

7406 

7641 

7875 

•2l3 

8110 

8344!  8578 

8812 

9046 

92791,  234 

186 

95i3 

9746 

9980 

•446 

•679  «9I2 

1 144 

i377 

1609  233 

3927|  232 

187 

271842 

2074 

23o6 

2538 

2770 

3ooi!  3233 

3464 

3696 

188 

•  41 58 

4389 

4620 

485o 

5o8i 

53 1 1  5542 

5772 

6002 

6232  23o 

189 

6462 

6692 
8982 

6921 

71 5i 

738o 

$a  zfJ 

8067 

8296 

8525j  229 
•806  228 

190 

278734 

9211 

943o 

9667 

•35i 

•578: 

191 

28io33 

1 26 1 

1488 

i7,5| 

1942 

2169!  23o6 
443 1  4606 

2622 

2849' 

3o75 

227 

192 

33oi 

3527 

3753 

3979i 

42o5 

4S82 

5107; 

5332 

226 

193 

5557 

5782 

6007 

6232i 

6456 

6681  6905 

7i3o 

7354 

7578 

2  23 

194 

7802 

8026 

8249 

8473; 

8696 

8920  9143 

9366 

9389 

9812 

223 

i95 

290035 

0257 

0480 

0702; 

0925 

1147I  i369 

i59i 

i8i3 

2o34 

222 

196 

2256 

2478 

2699 

29201 

3i4i 

33631  3584 

38o4 

4025 

4246 

221 

198 

4466 

4687 

4907 

5i27! 

5347 

5567!  5787 

6007!  62261 

6446 

220 

6665 

6884 

7104 

7323j 

7542 

7761  7979 

8198 

8ir6 

8635 

\\% 

199 

8853 

9071 

9289 

9)07 
1681 

9725 

99434.  »i6i 

•378 

•595| 

•8 1 3 

200 

3oio3o 

1247 

1464 

,898 
4009 

21 14'  233 1 

2547 

2764 

2980  217 

201 

3196 

3412 

3628 

3844J 

4275:  4491 

4706 

4921! 

5i36|  216 

202 

535i 

5566 

578i 

59Q6, 

621 1 

6425  6639 

6854 

7068! 

7282  2l5 

203 

7496 

7710 
9843 

7924 

8,37| 

835i 

8564!  8778 

8991 

9204 

94i7 

213 

204 

9630 

••56 

•268 

•481 

•693  #9o6 
2812  3o23 

,118 

i33o 

i542 

212 

2o5 

3i 1754 

1966 

2177 

2389 

2600 

3234 

3445, 

3656  211 

206 

3867 

4078 

4289 

4499 

4710 

4920  5i3o 

5340 

555i|  5760J  2io 

207 

5970  6 1 80 

63go 

65m 

8689; 

6809 

8898 

7018  7227 

7436 

7646 
973o 

209 

208 

8o63|  8272 

8481 

9106  9314 

9522 

9938;  208 

209 

3201461  o354 

o562 

0769 

0977 

1 184  i3oi 

3252  3438 

1 598 

i8o5 

2012  207 

210 

322219'  2426 

2633 

2839; 

3o46 

3665 

387! 

4077  206 

211 

4282:  4488 

4694 

4899 ! 

5io5 

53 10  55 1 6 

5721 

5926 

61 3 1  203 

212 

6336!  654i 

6745 

695o 

7i55 

735o  7563 
9398  9601 

7767 

7972 

8176  264 

213 

838o  8583 

87B7 

8991! 

9194 

98o5 

•••; 

•2  11   203 

214 

33o4i4'  0617 

0810 

1022 

1225 

i42-»  i63o 

1S32  2o34; 

2236 

215 

2438  26^0  2842 

3o44 

3246 

3447  364o 

4o5i 

4253  202 

2l6 

4454  4655  4856 

5o57 

0257 

5458  5658 

5859  6o5g 
7858  8o58 

6260  201 

217 

6460  6660  6860 

7060 

7260 

7459  7639 

8237  200 

218 

8456;  8656  8855 

9054 

9253 

945 1  9650!  9849  ••47 

•246  199 

2225  I98 

219 

340444  0642  0841 

1039 

1237 

1435!  1632!  i83o  2028 

N. 

0   1  1  |  2 

3   ! 

4 

5  |  6  |  7  1  8  | 

~v 

D. 

23 


4 

A  TABLE 

OF 

LOGARITHMS  FROM  1 

TO 

10,000. 

• 

N. 

0    I   I 

2 

3 

3oi4 

4 

3212 

5 

3409 

6 

7 

8  |  9 

D. 

197 

220 

342423  2620 

2817 

36o6 

38o2 

3999!  4196 
5962  6107 

221 

4392)  4589 
6353  6549 

4785 

4981 

5i78 

53741  5570  5766 
733o  7525  7720 

196 

222 

6744 

6939 

7i35 

7915  8110 
9860;  ••54 

i95 

223 

83o5  85oo 

8694 

8889 

9083 

927S  9472  9666 

194 

224 

350248 

0442 

o636 

0829 

1023 

1216 

1410 

i6o3 

1796;  1989 
3724!  3916 

i93 

225 

2i83 

2375 

2568 

2761 

2o54 

3 1 47 

3339 

3532 

i93 

226 

4108 

43oi 

4493 

4685 

4876 

5o68 

526o 

5452 

5643 1  5834 

192 

227 
228 

6026 

6217 

6408 

6099 

6790 

6981 
8886 

7172 

7363 

7554  7744 

191 

7935 

8125 

83i6 

85o6 

8696 

9076 

9266 

9*56 

9646 

ii 

229 

9835 

••25 

•2l5 

•404 

•593 

•783 

2S9 

1161 

i35o 

i539 

23o 

361728 

1917 

2105 

2294 

2482 

2671 

3o48 

3236 

3424 

23 1 

36i2 

3  800 

3988 

5862 

4176 

4363 

455i 

4739 

4926 

5n3 

53oi 

188 

232 

5488 

5675 

6049 
7915 

6236 

6423 

6610 

6796 
8659 
•5i3 

6983 

7169 

187 

233 

7356 

7542 

7729 

8101 

8287 

8473 

8845 

903  0 

186 

234 

9216 

94oi 

9087 

9772 

9958 
1806 

•i43 

•328 

•698 

•883 

1 85 

235 

371068 

1253 

i437 

1622 

3§?1 

2175 

236o 

2544 

2728 
4565 

184 

236 

2912 

3oo6 
49J2 

3280 

3464 

3647 

401 5 

4198 

4382 

184 

237 

4748 

5u5 

5298 

5481 

5664 

5846 

6029 

6212 

63g4 

1 83 

238 

6577 

6759 
858o 

6942 

7124 

73o6 

7483 

7670 

7852 

8o34 

8216 

182 

239 

8398 

8761 
0573 

8943 

9124 

93  06 

9487 

9668 

9849 

••3o 

181 

240 

38021 1 

0392 

0754 

2557 

0934 

1 1 1 5 

1296 

1476 

1 656 

i837 

181 

241 

2017 

2197 

2377 

2737 

2917 

3o97 

3277 

3456 

3636 

180 

242 

38i5 

3995 

5785 

4i74 

4353 

4533 

4712 

4891 

5070 
6856 

5249 

5428 

3 

243 

56o6 

5964 

6142 

6321 

6499 

6677 
8456 

7034 

7212 

244 

7390 

7568 

7746 

7923 
9698 

8101 

8279 

8634 

88n 

8989 

178 

245 

9166 

9343 

9520 

9875 

••5 1 

•228 

•4o5 

•532 

•759 

177 

246 

390935 

1 1 12 

1288 

1464 

1641 

1817 

1993 
375i 
55oi 

2169 

2345 

2521 

176 

247 

2697 
4452 

2873 

3048 

3224 

34oo 

3575 

3926 

4101 

4277 

176 

248 

4627 

4802 

4977 

5i52 

5326 

5676 

585o 

6025 

i75 

249 

6199 

6374 

6548 

6722 

6896 
8634 

7071 

7245 

7419 

7592 

7766 

95oi 

174 

25o 

397940 

8114 

8287 

8461 

8808 

8981 

9i54 

9328 

i73 

25l 

9674 

9847 

••20 

•192 

•365 

•538 

•711 

•883 

io56 

1228 

i73 

252 

401401 

i573 

H45 

1917 

2089 

2261 

2433 

26o5 

2777 

4663 

172 

253 

3m 

3292 

3464 

3635 

3807 

3978 

4149 

4320 

4492 

171 

254 

4834 

5oo5 

5i76 

5346 

55i7 

5688 

5858 

6029 

6199 

6370 

171 

255 

654o 

6710 

6881 

705 1 

8749 

7221 

739i 

756i 

773i 

9595 

1283 

8070 

X 

256 

8240 

8410 

8579 

8918 

9087 

9207 

9426 

9764 

257 

9933 

•102 

•271 

•44o 

•609 
229J 

•777 

•946 

1 1 14 

i45i 

$ 

258 

411620 

1788 

19D6 

2124 

2461 

2629 
43o5 

2796 

2964 

3i32 

259 

33oo 

3467 

3635 

38o3 

3970 

4i37 

4472 

4639 
63o8 

4806 

167 

260 

4U973 

5i4o 

53o7 

5474 

564i 

58o8 

5974 

6141 

6474 

167 

261 

6641 

6807 

&ll 

7i39 

73o6 

7472 

7638 

7804  7970 

8.35 

166 

262 

83oi 

8467 

8633 

8708 
•451 

8964 

9129 

9295 

9460 

9625 

9791 

i65 

263 

9956 

•121 

•286 

•616 

•781 

•945 
2^90 

1110 

1275 

1439 

i65 

264 

421604 

1788 

1933 

2097 

2261 

2426 

2754 

2918 

3o82 

1 64 

265 

3246 

3410 

3574 

3737 

3901 

4o65 

4228 

4392  4555 

4718 

164 

266 

4882 

5o45 

5208 

537i 

5o34 

5697 

586o 

6023'  6186 

6349 

1 63 

267 

65n 

6674 

6836 

6999 

7161 

7324 

7486 

7648,  781 1 

7973 

162 

268 

8i35 

8297 

845o 

8621 

8783;  8044 

9 1 06 

0268  9429 

959i 

162 

269 

9752 

9914 

••73 

•236 

•398  »559 

•720 

•881  j  1042 

1203 

161 

270 

43 1 364 

i525 

i685 

1846 

2007!  2167 

2328 

2488!  2649 

2809 

161 

271 

2969}  3i3o 

3290 

4888 

345o 

36io!  3770 

3930 
5525 

4090  4249  4409 

160 

272 

4569  4729 

5048 

5207!  5367 

5685  5844  6004 

1 59 

273 

6i63  6322 

6481 

6640  6798,  6957 
8226  83841  8542 

7.16 

7275  7433|  7592 

\U 

274 

775i  7909 

8067 

8701 

.  9017'  9175 

275 

9333  9491 

0648 

0806  0064  °I22 

•279 

•437:  »594  #752 

1 58 

276 

440909 ]  1 066 

1224  i38i  i533  1695 

i852 

2009!  2166!  2323 

1 57 

278 

2480J  2637 

27o3j  2g5o  3io6;  3263 
4357;  45i3  4669  4825 

3419 

3576!  3732;  3889 

1 57 

4o45 

4201 

4o3i 

5i37  5293  5449 

1 56 

279 

56o4 

5760 

59i5|  6071 

6226,  6382 

6537 

6692 j  6848J  7003 

1 55 
D. 

N. 

0 

1 

~T~I 

3  1 

4  |  5  |  6 

7  1  8  |  9 

A  TABLE 

OF 

LOGARITHMS  FROM  1  TO 

10,000. 

5 

N. 

0 

■ 

2 

3 

4 

5  |  6  |  7 

8 

8397 

9 

8552 

D. 
1 55 

280 

447158 

73i3 
8861 

7468 

7623 

7773 

7933  8oS8i  8242 

2S1 

8706 

90l5 

9170 

9324 

9478  9633  9787 

9941 

••95 

1 54 

282 

45o249 

o4o3 

6557 

07 II 

o865 

I0l8.  II72I  l320 

1479 

i633 

i54 

283 

1786 

1940 

2093 

2247 

2400 

2553  2706;  2859 

3012 

3i65 

1 53 

284 

33i8 

3471 

3624 

3777 
53o2 

3930 

4082  4235  4387 

4540 

4692 

1 53 

285 

4845 

4997 

5i5o 

5454 

56o6  57531  59io 

6062 

6214 

152 

286 

6366 

65i8 

6670 

6821 

6973 

7125  7276!  7428 

7579 

773i 

i5a 

2h 

7882 

8o33 

8184 

8336 

8487 

8633  8789  8940 

9091 

9242 

i5i 

288 

9392 

9543 

9694 

9845 

9995 

•1461  '296 j  »447 

•597 

•748 

i5i 

289 

460898 

1048 

1 198 

i348 

1499 

1649  '7991  1948 

2098 

2248 

i5o 

290 

462398 

2548 

2697 

2847 

2997 

3i46  3296  3445 

3594 

3744 

i5o 

291 

3893 
5383 

4042 

4191 

434o 

4490 

463g  4788;  4936 

5o85 

5234 

149 

292 

5532 

568o 

5829 

5977 

6126  6274'  6423 

657i 

6719 

3 

293 

6868 

7016 

7164 

73l2 

746o 

7608,  7756 

7904 

8o52 

8200 

294 

8347 

8495 

8643 

8790 

8933 

9o35!  9233 

933o 

9527 
•998 

9675 

148 

295 

9822 

9969 
U38 

•116 

•263 

•4 1 0 

•557;  «7o4 

•35 1 

1 U5 

147 

296 

471292 

1 585 

1732 

1878 

2025;  2171 

23i8 

2464 

2610 

146 

298 

27% 

2oo3 
4362 

3o4o 
45o8 

3i95 

334i 

3487  3633 

3779 

3925 
538i 

4071 

146 

4216 

4653 

till 

4944;  5090 

5235 

5526 

146 

299 

5671 

58i6 

5962 

6107 

6397J  6542 

6687 

6832 

6976 

145 

3oo 

477121 

7266 

74i  1 

7555 

lioo 

7844:  79s9 
9287  9  43 1 

8i33 

8278 

8422 

145 

3oi 

8566 

8711 

8855 

8999 

9M3 

9575 

9719 

9863 

144 

302 

480007 

oi5i 

0294 

o433 

o582 

0725  0869  1 01 2 

1 1 56 

1299 

144 

3o3 

1443 

1 586 

1729 

1872 

2016 

2 1 D9  23o2  2445 

2588 

2781 

143 

3  04 

2874 

3oi6 

3 1 09 

33o2 

3445 

3587 

373o,  3372 

40 1 5 

4157 

143 

3o5 

43  00 

4442 

4583 

4727 

4869 

5on 

5 1 53 ■  D295 

5437 

5579 

142 

3o6 

5721 

5863 

6oo5 

6147 

6289 

643o  6572  6714 

6855 

6997 

142 

307 

7i38 

7280 

7421 

7563 

7704 

7845 

79S6J  8127 

8269 

8410 

141 

3o8 

855i 

8692 

8S33 

8974 

9114 

9255 

9396  9537 

9677 

9818  141 

309 

9?58 

••99 

•239 

•33o 

•020 

•661 

•801!  °94' 

1081 

1222I  140 

3io 

49i362 

l502 

1642 

1782 

T9 

2062 

2201J  234i 

2481 

2621 

140 

3n 

2760 

2900 

3o4o 

3179 

3458 

35o7|  3737 
4989'  5i28 

3876 

4oi5 

i39 

3l2 

4i55 

4294 

4433 

4572 

471 1 

485o 

5267 

5406 

i39 

3i3 

5544 

5683 

5822 

5g6o 

6099 

7483 

6233 

6376,  65i5 

6653 

6791 

a 

3i4 

6930 

7068 

7206 

7344 

7621 

1V9\   7897 

8o35 

8i73 

3i5 

83n 

8448 

8586 

8724 

8862 

8999 

9i37i  9275 

9412 

955o 

1 38 

3i6 

9687 

9824 

9962 

#099 

•236 

•374 

•5ui  »648 

•785 

•922 

1 37 

3i7 

5oio59 

1 196 

1 333 

1470 

1607 

H4i 

1880J  2017 

2 1 54 

2291 
3655 

i37 

3i8 

2427 

2564 

2700 

2337 

2973 

3iog 

32461  3382 

35i8 

i36 

3 19 

379i 

3?27 

4o63 

4IQ9 

4335 

4471 

4607  j  4743 

4878 

5oi4 

1 36 

320 

oo5i5o 

5286 

5421 

5557 

5693 

5328 

5q64i  6099 
73 1 6  745i 

6234 

6370 

1 36 

i  321 

65o5 

6640 

6776 

691 1 

70  {6 

7181 

7586 

7721 

i35 

322 

7856 

799' 

8126 

8260 

8395 

853o 

8664  8799 

8934 

9068 

i35 

323 

9203 

9337 

9i7> 

9606 

9740 

9874 

•••9  »i43 

•27T 

•411 

i34 

324 

5ioD45 

0679 

08 1 3 

09 17 

10S1 

1 2 1 5 

1 349  1482 

1616 

1750 

1 34 

325 

1 883 

2017 

2l5l 

2284 

2418 

255i 

2684  2818 

2951 

3o84  1 33 

326 

32i8 

335i 

3484 

3617 

375o 

3883 

4016,  4149 

4282 

44i4  i33 

327 

4548 

4681 

48i3 

4940 

5o79 

521  1 

53441  5476 

5609 

574i 

i33 

328 

5874 

6006 

6i39 

6271 

64o3 

6535 

6668  6800 

6932 

7064 
8382 

132 

329 

7196 

7328 

746o 

7092 

7724 

7855 

7987;  8119 

825i 

1 32 

33o 

5i8oi4 

8616 

8777 

8009 

9040 

9171 

93o3  9434 

9566 

9697,  i3i 

33 1 

c  98^ 

9q5q 

••90 

•221 

•353 

•484 

•6i5 

•745 

•876 

1007  i3i 

332 

52ii38 

1269 

1400. 

i53o 

1661 

1792 

IQ22 

2o53 

2i83 

2814  i3i 

333 

2444 

2575 

2705 

2835 

2966 

3096  3226 

3356 

3486 

36 1 6,  i3o 

334 

3746 

3876 

4006 

4 1 36 

4266 

4396 

4526 

4656 

4785 

i3o 

335 

5o45 

5174 

53  04 

5434 

5563 

56o3 
6935 

5822 

59  5 1 

6081 

6210 

129 

336 

6339 

6469 

65q8 

6727 

6856 

71  14 

7243 

7372 

75oi 

129 

337 

763o 

7759 

733* 

8016 

8i45 

8274 

8402 

853 1 

8660 

8788 

!3 

338 

8917 

9045 

9'74 

93o2 

943o 

9759 

9687,  98l5 

9943 

••72 
i35i 

339 

53o2oo 

o328 

0456 

c584 

0712 

0840 

O968I  IO96 

1223 

128 

N. 

0 

1 

• 

3 

4 

5 

~6~|"T' 

~i 

9 

1>. 

3 

A  TABLE 

OF  LOGARITHMS 

3  FROM  1 

TO 

10,000. 

N. 
34o 

0 

1 1 2 

3 

4 

5 

6 

1 

8 

9 

D. 

128 

53U79 

1607!  1734 

1862 

1990 

2117 

2245 

2372 

2300 

2627 

34i 

2754 

2882'  3009 

3i36 

3264 

3391 

35i8 

3645 

3772 

3899 

127 

342 

4026 

4i53|  4280 

4407 

4534 

4661 

4787 

49'4 

5o4i 

D167 

127 

343 

5294 

542 1  j  5547 

D674 
6937 

58oo 

5927 

6o53 

6180 

63o6 

•  6432 

126 

344 

6558 

66b5  6811 

7063 

7189 

73 1 5 

744i 

7567 

7693 
895i 

126 

345 

7819 

7945  8071 

8197 
9462 

8322 

844s 

8574 

8699 

8823 

126 

346 

9076 

9202I  9327 

9578 

9703 

9829 

99^4 

••79 

•204 

125 

347 

540329 

0455 

0D80 

0705 

o83o 

0955 

1080 

1205 

i33o 

1454 

125 

348 

1579 

1704 

1829 

1953 

2078 

2203 

2327 

2452 

2576 

2701 

125 

349 

2825 

29D0 

3074 

3 199 

3323 

3447 

3571 

3696 

3820 

3944 

124 

35o 

544068 

4192 

43i6 

444o 

4564 

4688 

4812 

4936 

5o6o 

5i83 

124 

35i 

5307 

543i 

5555 

5678 

58o2 

5925 

6049 

6172 

6296 

6419 

124 

352 

6543 

6666 

6789 

6913 

7o36 

7i59 

7282 

74o5 
8635 

1529 
8758 

7652 
8881 

123 

353 

7775 

7898 

8021 

8144 

8267 

8389 

85i2 

123 

354 

9003 

9126 

9249 

0473 

937i 

9494 

9616 

9739 

9861 

9984 

•106 

123 

355 

550228 

o35i 

0095 

0717 

0840 

0962 

1084 

1206 

1328 

122 

356 

i45o 

1572 

1694 

1816 

1938 

2060 

2181 

23o3 

2425 

2547 

122 

357 

2668 

2790 

2911 

3o33 

3i55 

3276 

3398 

35i9 

364o 

3762 

121 

358 

3883 

4004 

4126 

4247 

4368 

4489 

4610 

473i 

4852 

49?3 
6182 

121 

359 

0094 

52i5 

5336 

5457 

5578 

5699 

5820 

594o 

6061 

121 

36o 

5563o3 

6423 

6544 

6664 

6785 

690D 

7026 

7i46 

7267 
8469 

7387 

120 

36i 

7507 

7627 

7748 

7868 

7988 

8108 

8228 

8349 

8589 

120 

36a 

8709 

8829 

8948 

9068 

9.88 

93o8 

9428 

9548 

9667 

9787 

120 

363 

9907 

••26 

•146 

•265 

•385 

•5o4 

•624 

•743 

•863 

•982 

119 

364 

56 1 1 01 

1221 

1 34o 

1459 

1 578 

1698 

2887 

1817 

i936 

2o55 

2174 

II9 

365 

2293 

2412 

253 1 

26D0 

2769 
3955 

3  006 

3i25 

3244 

3362 

II9 

366 

348i 

3  600 

37i8 

3837 

4074 

4192 

43n 

4429 

4548 

II9 

367 

4666 

4784 

4903 

5021 

5 1 39 

5257 

5376 

5494 

56i2 

573o 

110 

368 

5848 

5966 

6084 

6202 

6320 

6437 

6555 

6673 

6791 

6909 

Il8 

369 

7026 

7i44 

7262 

7379 

7497 

7614 
8788 

2$ 

7849 
9023 

7967 

8084 

Il8 

370 

568202 

83i9 

8436 

8554 

8671 

9140 

9257 

117 

i11 

9374 

9491 

9608 

9725 

9842 

9959 

••76 

•193 
1309 

2523 

•3og 

•426 

117 

372 

570543 

0660 

0776 

0893 

lOIO 

1126 

1243 

1476 

1592 

2755 

117 

373 

1709 

1825 

1942 

2o58 

2174 

2291 

3452 

2407 
3568 

2639 

Il6 

374 

2872 

2988 

3io4 

3220 

3336 

3684 

380o 

39i5 

Il6 

375 

4o3i 

4i47 

4263 

4379 

4494 
565o 

4610 

4726 

4841 

4957 

5072 

Il6 

376 

5 1 8b 

53o3 

5419 

5534 

5765 

5b8o 

5996 

6111 

6226 

ii5 

377 

634i 

6457 

6572 

6687 

6802 

6917 

7o3  2 

7147 

7262 
8410 

7377 

n5 

378 

7492 

7607 

7722 

7836 
8983 

795i 

8066 

8181 

8295 

8525 

iz5 

379 

8639 

8734 

8868 

9097 

9212 

9326 

9441 

9555 

9669 

114 

3bo 

579784 

9898 
1039 

••12 

•126 

•241 

•355 

•469 

•583 

•697 

•811 

114 

38i 

580925 

u53 

1267 

i38i 

i4g5 
263 1 

1608 

1722 

2858 

1 836 

1930 

114 

382 

2o63 

2177 

2291 

2404 

2Dl8 

2745 

2972 

3o85 

114 

383 

3199 
433 1 

33i2 

3426 

3539 

3652 

3765 

3879 

3992 

4io5 

4218 

n3 

384 

4444 

4557 

4670 

4783 

4896 

5009 

3122 

5235 

5348 

u3 

385 

5461 

5574 

5686 

5799 

5912 

6024 

6i37 

6230 

6362 

6475 

n3 

386 

6587 

6700 

6812 

6925 

7o37 

7149 

7262 

7374 
8496 

7486 

7399 

112 

387 

7711 

7823 

7935 

8047 

8160 

8272 

8384 

8608 

8720 

112 

388 

8832 

8944 

9o56 

9167 

9279 

9391 

95o3 

96l5 

9126 
•842 

9338 

112 

38g 

9950 

••61 

•i73 
1287 

•284 

•396 

•5o7 

•619 

•730 

•953 

112 

390 

591065 

1 176 
2288 

,399 

i5io 

1621 

i732 

2843 

1843 

1955 

2066 

in 

391 

2177 

23qq 

25lO 

262  r 

2732 

2954 

3o64 

3i75 

in 

392 

3286 

3397  35o8 

36i8 

3729 

3840 

3950 

4o6l 

4171 

4282 

in 

393 

43g3 

45o3  4614 

4724 

4834 

4945 

5o55 

5i65 

5276 

5386 

no 

394 

5496 

56o6  D7 1 7 

5827 

5937 

6047 

6 1 57 

6267 

6377 

6487 

no 

395 

6597 

6707  6817 

6927 

7o37 

7U6 

7256 

7366 

7476 
8372 

7586 

no 

396 

m 

78o5  7914 
8900:  9009 

8024 

8i34 

8243 

8353 

8462 

8681 

no 

397 

9119 

9228 

9337 

9446 

9  5  56 

9665 

9774 

109 

398 

9883 

9992  «I0I 

•210 

*3io 

•428 

•537 

•646 

•755 

•864 

109 

399 

600973 

1082  1191 

I299 

1408 

l5i7 

162D 

1734 

i843 

1931 

109 

N. 

0 

I 

2 

3 

4 

5 

6 

7 

* 

9 

D. 

A  TABLE 

OF  ] 

LOGARITHMS  FROM  1 

TO 

10,000. 

1 

N. 

.   0    |   I 

2 

3  |  4 

23861  2494 

2603 

6 

7  |  8 

9 

D. 

400 

6020601  2169 

2277 

2711 

28i9!  2928 
3902!  4010 

3o36 

108 

401 

3i44  3253 

336i 

3469!  3577 

3686 

3794 

4118 

108 

402 

4226 

4334 

4442 

455o 

4658 

4766 

4874 

4982:  5089 

5197 

108 

4o3 

53o5 

54i3 

552i 

5628 

5736 
63u 

5844 

5951 

6059'  6166 

6274 

108 

404 

638 1 

6489 

65g6 

6704 

6919 

7026 
8098 

7 i33 1  7241 
8205  83 1 2 

7348 

107 

4o5 

7455 

7062 

7669 
8740 

7777 
8847 

I884 

7991 

8419 

107 

406 

8526 

8633 

8954 

9061 

9167 

9274!  938i 

9488 

107 

407 

9594 

9701 

9808 

9914 

••21 

•128 

•234 

•34i 

•447 

•554 

107 

408 

6 1 0660 

0767 
1829 

0873 

0979!  1086 

1192 

2254 

129b 

i4o5 

i5n 

1617 

106 

409 

1723 

1936 

2042  2148 

236o 

2466 

2572 

2678 

106 

4io 

612784 

2890 

2996 

3l02 

3207 

33i3 

34>Q 
4475 

3525 

363o 

3736 

j  06 

4M 

3842 

3947 

4oD3 

4i59 
52i3 

4264 

4370 

458i 

4686 

4792 

106 

412 

4897 

5oo3 

5io8 

5319 

5424 

5529 

5634 

5740 

5845 

io5 

4i3 

5900 

6o55 

6160 

6265 

6370 

6476 

658i 

6686 

X 

6895 

io5 

4i4 

7000 

7io5 

7210 

73i5 

7420 

7525 
857i 

7629 

7734 

7943 
8989 

io5 

4i5 

8048 

8i53 

8257 

8362 

8466 

8676 

8780 

8884 

io5 

416 

9093 

9198 

9302 

9406 

95i  1 

9615 

9719 

9824 

9928 

••32 

104 

4i7 

62oi36 

0240 

o344 

0448 

o552 

o656 

0760 

0864 

0968 

1072 

104 

418 

1 176 

1280 

1 384 

1488 

1592 

i6o5 

1799 

1903 

2007 

2110 

104 

419 

*   22U 

23i8 

2421 

2525 

2628 

2732 

2835 

2939 
3973 

3o42 

3i46 

104 

420 

623249 

3353 

3456 

3559 

3663 

3766 

3869 

4076 

4i79 

io3 

421 

4282 

4385 

4488 

4591 

4695 

4798 

4901 

5oo4 

5107 
6i35 

5210 

io3 

422 

53i2 

54i  5 

55i8 

5621 

5724 

5827 

5929 

6o32 

6238 

io3 

423 

6340 

6443 

6546 

6648 

675i 

6853 

6g56 

7o58 
8082 

7161 

7263 

io3 

424 

7366 

7468 
8491 

757i 
8593 

7673 

7775 
8797 

7878 

7980 

8i85 

8287 

102 

425 

8389 

8695 

8900 

9002 

9104 

9206 

93o8 

102 

426 

9410 

9512 

9613 

9715 

9817 
o835 

9919 

••21 

•123 

•224 

•326 

102 

427 

63o428 

o53o 

o63i 

0733 

0936 

io38 

1 1 39 

1241 

1 342 

102 

428 

1444 

1 545 

1647 

1748 

1849 

1951 

2052 

2i53 

2255 

2356 

101 

429 

2457 

2559 

2660 

2761 

2862 

2963 

3064 

3i65 

3266 

3367 

101 

43o 

633468 

3569 

3670 

377i 

3872 

3973 

4074 

4175 

4276 

4376 

100 

43 1 

4477 

4578 

4679 

4779 
5785 

4880 

4981 

5o8i 

5i82 

5283 

5383 

100 

432 

5484 

5584 

568D 

5886 

5986 

6087 

6187 

6287 

6388 

100 

433 

6488 

6588 

6688 

6789 

6889 

6989 

7089 

7189 

7290 

7390 
8389 

100 

434 

7490 
8489 

7590 

7690 

7790 

7890 

7990 
8988 

8090 

8190 

9188 

8290 

99 

435 

8589 

8689 

8789 

8888 

9088 

9287 

9387 

99 

436 

9486 

9586 

9686 

9780 

9885 

9984 

••84 

•i83 

•283 

•382 

99 

437 

640481 

o58i 

068b 

0779 

0879 

0978 

1077 

1177 

1276 

i375 

99 

438 

J  474 

1573 

1672 

1771 

1871 

1970 

2069 
3o58 

2168 

2267 
3255 

2366 

99 

439 

2465 

2563 

2662 

2761 

2860 

2959 

3i56 

3354 

9 

44o 

643453 

355i 

365o 

3749 

3847 

3946 

4044 

4i43 

4242 

434o 

44 1 

4439 

4537 

4636 

4734 

4832 

493i 

5029 

5127 

5226 

5324 

98 

442 

5422 

552i 

5619 

5717 
6698 

58i5 

59i3 
6894 

601 1 

6110 

6208 

63o6 

98 

443 

6404 

65o2 

6600 

6796 

6992 

7089 
8067 

7187 
8i65 

7285 

9* 

444 

7383 

748i 

7579 
8555 

7676 

7774 

7872 

7969 
8945 

8262 

98 

445 

836o 

8458 

8653 

8750 

8848 

9043 

9140 

9237 

97 

446 

9335 

9432 

953o 

9627 

9724 

9821 

9919 

0890 

••16 

•n3 

•210 

97 

447 
448 

65o3o8 

o4o5 

o5o2 

0599 

0696 

0793 

0987 

1084 

1181 

97 

1278 

1375 

1472 

1569 

1666 

1762 

1809 

19D6 

2o53 

2i5o 

97 

449 

2246 

2343 

2440 

2536 

2633 

2730 

2826 

2923 
3888 

3019 

3n6 

u 

45o 

6532i3 

3309 

34o5 

35o2  3598 

36o5 

4658 

3791 

3984 

4080 

45 1 

4i77 

4273 

4369 

4465  4562 

47^4 

485o 

4946 

5o42 

96 

452 

5i38 

5235 

533 1 

5427 

5523 

5619 

57i5 

58io 

59o6 

6002 

96 

453 

6098 
7006 

6194 

6290 

6386 

6482 

6577 

6673 

6769  6864 
7726!  7820 
8679'  8774 

6960 

96 

454 

7i52 

7247 

7343  7438 

7534 

1629 

8584 

S70 

96 

455 

891 1 

8107 

8202 

8298:  8393 

8488 

95  1 

456 

8965 

9060 

91 55 

925o  9346 

9441 

9536 

963i|  9726 

9821 

95 

457 

9916;  *#i 1 

•106 

•2011  ^296 

•3qi 

•486 

•58i  »676 

•771 

90 

458 

66o865  0960 

io55 

n5o!  1245 

i339 

1434 

i52o'  i623 
2476  2069 

1718 

95 

459 

N. 

i8i3 

1907 

2002 

2096,  2191 

2286 

238o 

2663 

95 

0 

I 

» 

3  |  4 

5 

6 

7 

8 

9 

D. 

23* 


A    TABLE    OF    LOGARITHMS    FROM    1    TO    10,000. 


N. 

0 

I 

2 

;  3 

I  4  |  5 

6  |  7 

8 

Ll 

D. 

460 

662758 

2852 

2947 

3o4i 

3i35j  323o  3324!  34i8 

35i2 

3607 

94 

461 

3701 
4642 

3703 

4736 

3»89 

3983 

4078:  4172:  4266'  436o 

4454 

:  4548 

94 

462 

483o 

4924 

5oi8'  5i  12  52o6!  5299 

5393;  5487 

94 

463 

558i 

5675 

5769 
6705 

5862 

5956j  6o5ol  6143!  6237 

633 1 

6424 

94 

464 

65i8 

6612 

6799 

1  6892!  6986  7079  7173 

7266 

736o 

94 

465 

7453 
8386 

7546 

7640 

7733 
8665 

7826  7920!  8oi3 
8759!  8852  8945 

8106 

8199 

8293 

93 

466 

8479 

8572 

9o38 

9i3i 

9224 

93 

467 

c   93lI 

9410 

95o3 

9596 

9689;  9782 

9875 

9967 

••60 

•i53 

93 

468 

670246 

o33g 

o43i 

0D24 

0617 

0710 

0802 

0895 

0988 

1080 

93 

469 

i.73 

1265 

i358 

i45i 

1 543 

1636 

1728 

1821 

1913 

2005 

93 

470 

672098 

2190 

2283 

2375 

2467 

256o  2652 

2744 

2836 

2929 

92 

4?i 

3o2i 

3n3 

32o5 

3297 

3390 

3482 

3574 

3666 

3758 

385o 

92 

472 

3942 

4o34 

4126 

4218 

43 10 

4402 

4494 

4586 

4677 

4769 

92 

473 

4861 

4953 

5o45 

5i37 

5228 

5320 

5412 

55o3 

5595 

5687 

92 

474 

5778 

5870 

5962 

6o53 

6i45 

6236 

6328 

6419 
7333 

65n 

6602 

92 

475 

6694 

6785 

6876 
7789 

6968 

7o59 

7.5! 

7242 

'  7424 

75i6 

91 

476 

7607 

1698 
8609 

7881 
879 1 

7972 

8o63 

81 54 

8245 

8336 

8427 

9* 

477 
478 

85i8 

8700 

8882 

8973 
9882 

9064 

91 55 

9246 

9337 

91 

9428 

95i9 

9610 

9700 

979  « 

9973 

••63 

•i54 

•245 

91 

479 

68o336 

0426 

o5n 

0607 

0698 

0789 

0879 

0970 

1060 

1  i5i 

9i 

480 

681241 

i332 

1422 

i5i3 

i6o3 

1693 

1784 

1874 

1964 

2o55 

90 

481 

2i45 

2235 

2326 

2416 

25o6 

2596 

2686 

2777 

2867 

2o57 

3857 

90 

482 

3o47 

3i37 

3227 

33i7 

3407 

3497 

3587 

3677 

3767 

90 

483 

3947 

4o37 

4127 

4217 

43o7 

4396 

4486 

4576 

4666 

4756 

90 

484 

4845 

4935 

5o?5 

5i  14 

D204 

5294 

5383 

5473 

5563 

5652 

00 

485 

5742 

583 1 

5921 

6010 

6100 

6189 

6279 

6368 

6458 

6547 

°9 

486 

6636 

6726 

68 1 5 

6904 

6994 

7o83 

7172 

7261 

735i 

744o 

^ 

487 

7529 

7618 

7707 

7796 

7886 

7975 

8064 

8i53 

8242 

833 1 

89 

488 

8420 

8509 

85o8 

8687 

8776 

8865 

8953 

9042 

91 3 1 

9220 

o9 

489 

9309 

93o8 
0285 

9486 

9575 

9664 

9753 

9841 

9o3o 
0816 

-10 

0905 

•107 

& 

490 

690196 

o373 

0462 

o55o 

0639 

0728 

0993 

h 

491 

1081 

1170 

1258 

i347 

1435 

i524 

1612 

1700 
2583 

1789 

1877 

88 

492 

1965 

2o53 

2142 

2230 

23i8 

2406 

2494 

2671 

2759 

88 

493 

2847 

2q35 

3o23 

3 1 1 1 

3 1 99 

3287 

3375 

3463 

355i 

3639 

88 

494 

3727 

38i5 

3r;o3 

3991 

4078 

4166 

4254 

4342 

443o 

45i7 

88 

495 

46o5 

4693 

4781 

4868 

4956 

5o44 

5i3i 

5219 

53o7 

5394 

88 

496 

5482 

5569 

5657 

5744 

5832 

59i9 

6007 

6094 

6182 

6269 

!? 

497 

6356 

6444 

653 1 

6618 

6706 

•6793 

6880 

6968 

7839 
8709 

7o55 

7142 

87 

498 

7229 

7317 

74o4 

7491 

7578 
8449 

7665 

7752 

7926 

8014 

H7 

499 

8101 

8188 

8275 

8362 

8535 

8622 

8796 

8883 

57 

5oo 

698970 

9057 

9144 

923 1 

9317 

9404 

9I91 

9578 

9664 

975i 

2? 

5oi 

9«38 

9924 

••11 

••98 

•184 

•271 

•358 

•444 

•53 1 

•617 

ll 

502 

700704 

0790 

0877 

0963 

io5o 

n36 

1222 

1 309 

i395 

1482 

5o3 

1 568 

1654 

I74i 

1827 

1913 

1099 
2861 

2086 

2172 

2258 

2344 

86 

5o4 

243 1 

2517 

26o3 

2689 

2775 

29  47 

3o33 

3i  19 

32o5 

86 

5o5 

3291 

3377 

3463 

3549 

3635 

3721 

3807 

38o3 

4731 

3979 

4o65 

86 

5o6 

4101 

4236 

4322 

44o8  4494 

4579 

4665 

4837 

4922 

86 

507 

5oo8 

5094 

5 179 

5265  535o 

5436 

5522 

5607 

56q3 

5778 

86 

5o8 

5864 

5949 
68o3 

6o35 

6120  6206 

6291 

6376 

6462' 

6547 

6632 

85 

509 

6718 

6888 

6974  7059 

7'44 

7229 

8081 

73i5| 

74oo 

7485 

85 

5io 

707570 
8421 

7655 

7740 
859i 

7826  79 1 1 

7996 

8166 

825i 

8336 

85 

•MI 

85o6 

8676  8761 

8846 

8931 

9015 

9100 

9?85 

85 

5l2 

9270 

9355 

944o 

9524  9609 

9694 

9779 

9863 

9948  j 

••33 

85 

5i3 

710117 

0202 

0287, 

0371  0456  o54o 

0625 

0710 

0794  J 
16391 

0879 

85 

5i4 

0963  (  1048 
18071  1802 
265o  2734 

Il32 

1217  i3or  i385 

1470 

1 5  54 

1723 

84 

5i5 

1976J 
2818, 

2o6o!  2144  2229 

23i3 

2397.  2481]  2J66 

84 

5i6 

2902  29^6  3j7o 
37^2  3826  39io 

3 1 54 

3238  3323  3407 

84 

5i7 

3491  3575 

3659 

3  99  4 

4078J  4i62!  42  .6 

84 

5i8 

433o  4414 

4081  4665  4749  4833 

4916  5ooo  .)o84 

84 

5.9 

5i67 

D2DI 

5335 

54i8 

55o2  5586,  5669 

5753 j  5836j  5920 

_8i_ 
D. 

N. 

0 

I 

2 

3 

4 

5  |  6 

7  |  8  1  9 

A    TABLE    OF    LOGARITHMS    FROM    1    TO    10,000. 


N. 

0 

■ 

2 

3 

4 

5 

6  J  7 

8 

9 

D. 

520 

716003 

6087 

6170 

6254 

6337 

6421 

65o4 

6588 

6671 

6754 

83 

521 

SS38 

6921 

7004 

7088 

1171 
8oo3 

7254 

7338 

lA2\ 

75o4 
8336 

7587 

83 

522 

7671 

7754 

7837 

7920 

8086 

8169 

8253 

8419 

83 

523 

85o2 

8585 

8668 

875i 

8834 

8917 

9000  9083 

9i65 

9248 

83 

524 

933i 

9414 

9497 

958o 

9663 

9745 

9828  991 1 

9994 

•#77 

83 

523 

720159 

0242 

0323 

0407 1  0490 

o573 

o655|  0738 

0821 

0903 

83 

526 

0986 

1068 

ii5i 

1233 

i3i6 

1 398 

1481 

1 563 

1646 

1728 

82 

527 

1811 

1893 

1975 

2o58 

2140 

2222 

23o5 

2387 

2469  2552 

S2 

528 

2634 

2716 

2798 

2881 

2963 
3784 

3o45 

3127 

3948 

3209 

3291 

3374 

82  . 

529 

3456 

3538 

3620 

3702 

3866 

4o3o 

4112 

4194 

82 

53o 

724276 

4358 

4440 

4522 

4604 

4685 

4767 

4849 

4931 

5oi3 

82 

53i 

5093 

5i76 

5258 

5340 

5422 

55o3 

5585 

5667 

5748 

583o 

82 

532 

5912 

5993 

6075 

6i56 

6238 

632o 

6401 

6483 

6564 

6646 

82 

533 

6727 

6809 
7623 

6890 

6972 

7o53 

7i34 

7216 

7297 

7379 

7460 
8273 

81 

534 

734i 

7704 

7785 

7866 

7948 

8029 

8110 

8191 

81 

535 

8354 

8435 

85i6 

8597 

8678 

8759 

8841 

8922 

9003 

9084 

81 

536 

9165 

9246 

9327 

9408 

9489 

957o 

965i 

9732 

98i3 

9893 

81 

^ 

9974 

730782 

®»55 

•i36 

•217 

•298 

•378 

•459  °54o 

•621 

•702 

81 

538 

o863 

0944 

1024 

no5 

11S6 

1266 

1 347 

1428 

i5o8 

81 

539 

1589 

1669 

i75o 

i83o 

1911 

1991 

2072 

2l52 

2233 

23i3 

81 

54o 

732394 

2474 

2555 

2635 

2715 

2796 

2876 

2g56 

3o37 

3117 

80 

54i 

3197 

3278 

3358 

3438 

35i8 

3598 

3679 

3759 

3839 

3919 

80 

542 

3999 

4079 

4160 

4240 

4320 

44oo 

4480 

456o 

4640 

4720 

80 

543 

4800 

4880 

4960 

5o4o 

5l20 

5200 

5270 

6078 

5359 

5439 

55i9 

80 

544 

5599 

5679 

5759 

5838 

59i8 

5998 

6i57 

6237 

63i7 

80 

545 

6397 

6476 

6556 

6635 

67i5 

6795 
7590 

8384 

6874 

6954 

7o34 

71 13 

80 

546 

7193 

7272 

7352 

743i 

75n 

7670 

7749 

7829 

7908 

79 

547 

7987 

8067 

8146 

8225 

83o5 

8463 

8543 

8622 

8701 

79 

548 

9572 

8860 

8939 

9018 

9097 

9177 

9256 

9335 

9414 

9493 

79 

549 

9651 

973i 

9810 

9889 
0678 

9968 

••47 

•126 

•205 

•284 

79 

55o 

74o363 

0442 

0521 

0600 

0757 

o836 

0915 

0994 

1782 

1073 

79 

55i 

Il52 

1230 

i3o9 

1 388 

1467 

i546 

1624 

1703 

i860 

79 

552 

i939 

2018 

2096 

2175 

2254 

2332 

241 1 

2489 

2568 

2647 

71 

553 

2720 

2804 

2882 

2961 

3o39 
3823 

3u8 

3 196 
398o 

327D 

3353 

343 1 

554 

35io 

3588 

3667 

3745 
4528 

3902 

4o58 

4i36 

42i5 

78 

555 

4293 

4371 

4449 

4606 

4684 

4762 

4840 

4919 

4997 

78 

556 

5075 

5i  53 

523i 

53o9 

5387 

5465 

5543 

562i 

5699 

5777 

78 

557 

5855 

5933 

601 1 

6089 

6167 

6245 

6323 

6401 

6479 

6556 

78 

558 

6634 

6712 

6790 

6.868 

6945 

7023 

7101 

7955 

7256 

7334 

78 

559 

74i2 

7489 

7567 

7643 
8421 

7722 

7800 

7878 

8o33 

8110 

78 

56o 

748188 

8266 

8343 

8498 

8576 

8653 

8731 

880S 

8885 

77 

56i 

8963 

9040 

91 18 

919'j 

9272 

935o 

9427 

95o4 

9582 

9659 

77 

562 

9736 

9«»4 

9891 

9968 

©•45 

•i23 

•200 

•277 

•354 

•43 1 

77 

563 

75o5o8 

o586 

0663 

0740 

0817 

0894 

0971 

1048 

I  1  25 

1202 

77 

564 

1279 

1 356 

1433 

i5io 

1 587 

1664 

1 741 

1818 

1895 

1972 

77 

565 

2048 

2125 

2202 

2279 

2356 

2433 

2509 

2586 

2663 

2740 

77 

566 

2816 

2893 

297O 

3  047 

3i23 

3200 

3277 

3353 

343o 

35o6 

77 

£z 

3583 

366o 

3736 

38i3 

3889 

3966 

4042 

4i  19 

4883 

4195 

4272 

77 

568 

4348 

4425 

45oi 

4578 

4654 

4730 

4807 

4960 

5o36 

76 

569 

5lI2 

5189 

5265 

534i 

54i7 

5494 

6256 

5570 

5646 

5722 

5799 

76 

570 

755875 

595i 

6027 

6io3 

6180 

6332 

6408 

6484 

656o 

76 

57 1 

6636 

6712 

6788 

6864 

6940 

7016 

]Z 

7168 

7244 

7320 
8079 

76 

572 

7396 

7472 

7548 
83o6 

7624 
8382 

7700 

7775 

7927 

8oo3 

76 

573 

8i55 

8230 

8458 

8533 

8609 

8685 

8761 
9517 

8836 

76 

574 

8912 

8988 

9o63 

9139 

9214 

9290 

9366 

944i 

9592 

76 

575 

9668 

9743 

9819 

9970 

••45 

•121 

•196 

•272 

•347 

73 

1  57° 

760422 

0498 

0J73 

0649 

O724 

0799 
i552 

o875 

0950 

I025 

1101 

75 

578 

1176 

I25l 

1326 

1402 

1477 
2228 

1627 

1702 

1778 

1 853 

75 

1928 

2oo3 

2078 

2i53 

23o3 

2378 

2453 

2529 

3278 

2604 

75 

579 

2679 

2754 

2829 

2904 

2978 

3o53 

3i28 

32o3 

3353 

75 

N. 

0 

I 

a 

3 

4 

5  |  6 

7 

8    9 

I). 

10 


A    TABLE    OF    LOGARITHMS    FROM    1    TO    10,000. 


N. 
~58o~ 

0 

1 

2 

1  3 
3653 

4 

3727 

5 

38o2 

6  |  7  |  8  1  9 

D. 

763428  35o3 

3578 

3877I  3952  4027!  4101 

75 

58 1 

4176;  425i 

4326 

1  44oo 

4475 

455o 

4624  4699!  4774 
5370!  5445J  5520 

4848 

75 

582 

4923i  4998|  5072 
5669!  5743  58 1 8 

1  5i47 

j  5221 

5296 

5594 
6338 

75 

583 

5892 

'    5966 

6041 

61 i5  6iqo;  6264 

74 

584 

64i3 

6487 

6562 

6636 

1  67IO 

6785 

6859 

6933[  7007 

7082 

74 

585 

7i56 

723o 

7304 

7379 
8120 

7453 

i  8l94 
,-  8934 

7527 

7601 

7675;  7749 

7823 

74 

586 

7898 
8638 

7972 

8046 

8268 

8342 

841 6|  84905  8564 
9i56!  923oi  93o3 

74 

587 
588 

8712 

8786 

8860 

9008 

9082 

74 

9377 
770115 

945i 

9625 

9599 

|  9673 

9746 

9820 

9894  9968J  ••42 

74 

589 

oi89 

0263 

o336 

0410 

0484 

o557 

o63ij  0705 

0778 

74 

59o 

770832 

0926 

0999 
1734 

1073 

1 146 

1220 

1293 

1367  1440 

i5i4 

74 

59i 

1587 

1661 

1808 

1881 

1955 

2028 

2102,  2175 

2248 

73 

5o2 

2322 

2395 

2468 

2542 

26i5 

2688 

2762 

2835  2908 

2981 

73 

593 

3o55 

3i28 

3201 

3274 

3348 

342i 

3494 

3567;  364o 

37i3 

73 

594 

3786 

386o 

3933 

4006 

4079 

41 52 

4225 

4298  4371 

4444 

73 

595 

45i7 

459o 

4663 

4736 

4809 
5538 

4882 

4955 

5o28 

5 1 00 

5i73 

73 

596 

5246 

53i9 

5392 

5465 

56io 

5683 

5756 

5829 

5902 

73 

P 

5974 

6047 

6120 

6193 

6265 

6338 

641 1 

6483 

6556 

6629 

73 

6701 

6774 

6846 

6919 

6992 

7064 

7137 

7209!  7282 

7354 
8079 

73 

599 

7427 
778i5i 

7499 

7572 

7644 
8368 

77n 

7789 

7862 
8585 

79341  8006 

72 

ooo 

8224 

8296 

8441 

85i3 

8658  8730 

8802 

72 

6oi 

8874 

8947 

9019 

9091 

9i63 

9236 

93o8 

938o  9452 

9524 

72 

602 

9396 

9669 

974i 

9813 

9885 

9957 

••29 

•101!  »i-]3 

•245 

72 

6o3 

780317 

o389 

0461 

o533 

o6o5 

0677 

0749 

1468 

0821  o893 

0965 

72 

6o4 

1037 

1 1 09 

ii8j 

1253 

i324 

1 396 

i54o  1612 

1684 

72 

6o5 

1755 

1827 

1899 

1971 

2042 

2114 

2186 

2258  2329 

2401 

72 

606 

2473 

2544 

2616 

2688 

275o 
3473 
4189 

283 1 

2902 

2974  3046 
3689!  3761 

3 1 1 7 

72 

607 
608 

3i89 

3260 

3332 

34o3 

3546 

36i8 

38321  71 

39o4 

3975 

4046 

4118 

4261 

4332 

44o3 

4475 

4546 

7i 

6o9 

4617 

4689 

4760 

483 1 

4902 

4974 

5o45 

5n6 

5i87 

5259 

7i 

610 

78533o 

54oi 

5472 

5543 

56i5 

5686 

5757 

5828 

5899 

5970 

71 

611 

6041 

6112 

6i83 

6254 

6325 

6396 

6467 

6538 

6609 

6680 

71 

612 

675i 

6822 

6893 

6964 

7o35 

7106 

7177 

7248 

73i9 

739o 

7i 

6i3 

746o 
8168 

753i 

7602 

7673 
838i 

7744 

78i5 

7885 

7956 
8663 

8027 

8098 

7i 

614 

8239 

83  ro 

845i 

8522 

85g3 

8734 

8804 

7i 

6i5 

8875 

8946 

9016 

9087 

9i57 

9228 

9299 

9369 

9440 

95io 

7i 

616 

958i  965i 

9722 

9792 

9863 

9933 

•••4 

••74 

•i44 

•2l5 

70 

6^8 

7902S5  o356 

0426 

0496 

0567 

0637 

0707 

0778 

0848 

0918 

70 

0988 

1009 

1129 

1199 

1269 

1 34o 

1410 

1480 

i55o 

1620 

70 

6i9 

1691 

1761 

i83i 

1901 

1971 

2041 

2 1 1 1 

2181 

2252 

2322 

70 

620 

792392 

2462 

2532 

2602 

2672 

2742 

2812 

2882 

2952 

3022 

70 

621 

3092 

3i62 

323i 

33oi 

337i 

3441 

35n 

358i 

365i 

3721 

70 

622 

379o 

386o 

3930 

4000 

4070 

4i39 

4209 

4279 

4349 

44l8 

70 

623 

4488 

4558 

4627 

4697 

4767 

4836 

4906 

4976 

5o4D 

5u5 

70 

624 

5i85''  5254 

5324 

53  9  3 
6088 

5463 

5532  56o2 

5672 

5741 

58n 

70 

625 

588o|  5949 

6019 

67i3 

6i58 

6227 

6297 

6366 

6436 

65o5 

69 

626 

6374'  6644 

6782 

6852 

692 1 
7614 
83o5 

6990 

7060 

7129 

7198 

69 

627 

7268!  7337 
7960:  8029 
865i|  8720 

74o6 

7475 

7545 

7683 
8374 

7752 

7821 
85i3 

7890 
8582 

69 

628 

8098 

8167 

8236 

8443 

69 

^9 

8780 
947» 

8858 

8927 

8996 

9065 

9i34 

92o3 

9272 

69 

63o 

799341  9400 
800029!  0098 

9547 

9616 

9685: 

9754 

9823 

9892 

9961 

69 

63i  | 

0167 

0236 

o3o5 

0373 

0442 

o5u 

o58o 

0648 

69 

632  | 

0717  0786 

o854 

0923 

0992! 

1 06 1 

1 1 29 
i8i5 

1 198 

1884 

1266 

i335 

69 

633 

1404  1472 

i:')ii 

1 609 
2295 

1678! 

»747| 

iq52 

2021 

69 

634  i 

2089  2 1 58 

2226 

23631 

24J2 

25oo 

2568 

2637 

27o5 

3 

635  ! 

2774  2842 

2910: 

2979 

3o^7 

3 116 

3 1 84 

3252  332i ! 

3389 

636  i 

3457  3525 

3594: 

3662 

373o!  3798  3867 
4412  448o  4548 

3935 

400  3 

4071 

68 

637  ! 

4i39  4208 

4276 

4341 

4616 

4685  4753, 

68 

638 

4821  4889 

4957 1 

5o25 

5o93j  5i6i 

5229 
59o8 

5297 

5365 

5433 

68 

639 

55oi|  5569 

5637,  57o5j 

5773I  584i 

5976 

6044 

6112 

68 

N.  | 

0   |  1 

2  J  3  | 

4  1  5 

6 

7 

8 

9 

D. 

A-  TABLE 

OF  LOGARITHMS  FROM  1 

TO 

10,000. 

11 

N. 
640 

0 

>  1 ' 

3 

4 

645 1 

5 
65i9 

6 

' 

8 

"J 

D. 

806180 

6248!  63i6 

6384 

6587 

6655 

6723 

6790 

"68~ 

641 

6858 

6926  6994 

7061 

7129 

7197 

7264 

7332 

7400 

7467:  68 

642 

7535 

7603!  7670 

7733 

7806 

7873 

794? 

8008 

8076 

8i43i  68 

643 

8211 

8279!  8346 

8414 

8481 

8549 

8616 

8684 

875i 

8818  67 

644 

8886 

8953 

9021 

9088 

9 1 56 

9223 

9290 

9358 

9425 

9492 

67 

645 

9560 

9627 

9694 

9762 

9829 

9896 

9964 

••3 1 

••98 

•i65 

67 

646 

8i0233 

o3oo 

o3&7 

0434 

o5oi 

0569 

o636 

0703 

0770 

0837 

67 

647 

0904 

0971 

1039 

1 1 06 

1 173 

1240 

i3o7 

1374 

i44i 

i5o8 

i1 

648 

1075 

1642 

1709 

1776 

i843 

1910 

1977 

2044 

2111 

2178 

67 

649 

2245 

23l2 

2379 

2445 

2DI2 

2579 

2646 

2713 

2780 

2847 

67 

65o 

8i29i3 

2980 

3o47 

3i  14 

3i8i 

3247 

33i4 

338i 

3448 

35i4 

67 

65 1 

358i 

3648 

3714 

378i 

3848 

3914 
458i 

3981 

4048 

4ii4 

4181 

67 

652 

4248 

43U 

438i 

4447 

45i4 

4647 

47U 

4780 

4847 

67 

653 

49i3 

4980 

5o46 

5u3 

5179 

5246 

53i2 

5378 

5445 

55u 

66 

654 

5578 

5644 

571 1 

5777 

5843 

5oio 
6573 

5976 

6042 

6109 

6175 

66 

655 

6241 

63o8 

6374 
7o36 

644o 

65o6 

663g 

6705 

6771 

6838 

66 

656 

6904 

6970 

7102 

7169 

7235 

73oi 

7367 

7433 

7499 

66 

657 

7565 

763 1 

&S 

7764 

7830 

7896 
8556 

7962 

8028 

8094 

8160 

66 

658 

8226 

8292 
895i 

8424 

8490 

8622 

8688 

8754 

8820 

66 

659 

8885 

9017 

9083 

9149  92i5 

9281 

9346 

9412 

9478 

66 

660 

8i9544 

9610 

9676 

974i 

9807  9873 

9939 

•••4 

••70 

•i36 

66 

661 

820201 

0267 

o333 

0399 
io55 

0464I  o53o 

0593!  0661 

0727 

0792 

66 

662 

o858 

0924 
1579 

09S9 

1120  1186 

1231 

i3 17 

i382 

1448 

66 

663 

i5i4 

1645 

1710 

1775  1841 

1906 

1972 

2037 

2103 

65 

664 

2168 

2233 

2299 
2962 

2364 

243o  2495 

2  360 

2626 

2691 

2756 

65 

665 

2822 

2887 

3oi8 

3o83  3i48 

32i3 

3279 

3344 

3409 

65 

666 

3474 

3539 

•  36o5 

3670 

3735  38oo 

3865 

3930 

3996 

4061 

65 

667 

4126 

4I91 

4256 

432! 

4386 1  445 1 

45i6 

458 1 

4646 

47 1 1 

65 

668 

4776 

-4841 

4906 

4971 

5o36 

5ioi 

5i66 

523i 

5296 

536 1 

65 

669 

5426 

549  > 

5556 

562i 

5686 

5751 

58i5 

588o 

5945 
6593 

6010 

65 

670 

826075 

6140 

6204 

6269 
6917 

6334 

6399 

6464 

6528 

6658 

65 

671 

6723 

6787 

6852 

6981 

7046 

7111 

7175 

7240 

73o5^  65 

2s 

7369 
801 5 

7434 

7499 

7563 

7628  7692 

7737 

7821 

7886 
853 1 

795 1 

65 

673 

8080 

8i44 

8209 

8273 !  8333 

8402 

8467 

85o5 

9239 

64 

674 

8660 

8724 

8789 

8853 

8918!  8982 

9046 

9111 

9173 

64 

675 

93o4 

9368 

9432 

9497 

9561!  9625 

9690;  9754 

9818 

9882 

64 

676 

9947 

••11 

•a75 

•i39 

•204I  *268 

«332'  «396 

•460 

•525 

64 

677 

830389 

o653j  0717 

0781 

o845[  0909 
1486  i55o 

0973:  1037 

1102 

1 166 

64 

678 

1230 

1204 
1934 

i358 

1422 

1614  1678 

1742 

1806 

64 

679 

1870 

1958 
2637 

2062 

2126:  2189 
2764'  2828 

2253,  2317 

238i 

2445 

64 

680 

832509 

2373 

2700 

2892!  2956 

3020 

3o83 

64 

681 

3i47 

3211 

3275 
3912 

3338 

3402  3466 

353o!  35g3 

3657 

372i 

64 

682 

3784 

3848 

3975 

4039!  4io3 

4106'  423o 

4294 

4357 

64 

683 

4421 

4484 

4348 

461 1 

4673;  473q 

4802  4866 

4929 

4993 

64 

684 

5o56 

5l20 

5i83 

5247 

53io!  5373 

5437  55oo 

5564 

5627 

63 

685 

5691 

5754 

58i7 

588 1 

5g44  6007 
6577  66/*i 

6071  61 34 

6197 

6261 

63 

686 

6324 

6337 

645i 

65i4 

6704!  6767 

683o 

6894 

63 

687 

6957 
7288 

7020 

7o83 

7146 

7210,  7273 

7336  7399 

7462 

7525 

63 

688 

7632 

77i5 

7778 

7841  7?°4 
8471!  8334 

7967  8o3o 

8o93 

81 56 

63 

689 

8219 

8282 

8345 

8408 

85g7  8660 

8723 

8786 

63 

690 

838849 

8912 

8975 

9038 

9101  9164 

9227  9289 

9352 

94i  5 

63 

691 

9478 

9341 

9604 

9667 

9729  9792 

q855  9918 

998i 

••43'  63 

692 

840106 

0169 

0232 

0294 

o357  0420 

0482  o543 

0608 

0671;  63 

693 

0733 

0796 

0859 

0921 

0984  1046 

1109  1172 

1234 

1297J  63 

694 

1 3  39 
1985 

1422 

1485 

1347 

1610  1672 

1-35  1797 

i860 

1922:  63 

693 

20  '.7 

21 10 

2172 

2235  2297 

236o  2422 

2484 

2347   62 

696 

2609 

2672 

2734 

2796 

2839  2921 

2933  3o46 

3io8 

3 1 70  62 

677 

3233 

3295 

3357 

3 ',20 

3482  3344 

36o6  »3669 

373i 

3793  62 

698 

3855 

3ni8 
4639 

398o 

4042 

4104  4166 

4229  4291 

4353 

44i 5   62 

699 

N. 

4477 

4601 

4664 

4726  4788 

485o,  4912 

4974 

5o36'  62 

0 

I 

2 

3 

4    5 

6  I  7 

8 

9  Id. 

12 

A  TABLE 

OF 

LOGARITHMS  FROM  1 

TO 

10,0C 

0. 

N. 

0 

X   |   2 

3  |  4  |  s  | 

6 

7 

8  1  ' 

62 

700 

845098,  5l6o  0222 

57i8  578o|  5842 

5284  5346|  5408 

D470 

5532 

5594 

5656 

701 

5904  5966  602S 
6523  6585  6646 

6090 

6i5i 

62i3 

6275 

62 

702 

6337  6399!  6461 

6708 

6770 

6832 

6894 

62 

703 

69D5  7017 

7079 

7141 

7202  7264 

7326 

7388 

7449 

75n 

62 

704 

7573 

7634 

7696 

7758 

7819 

7881 
8497 

7943 
8559 

8004 

8066 

8128 

62 

7o5 

8189 

825i 

83i2 

8374 
8989 

8433 

8620 

8682 

8743 

62 

706 

88o5 

8866 

8928 

9o5i 

91 12 

9174 

9235 

9297 

9358 

61 

708 

9419 

948i 

9542 

9604 

9665 

9726 

9788 

9849 

9011 

9972 

61 

85oo33 

0095 

01 56 

0217 

0279 

o34o 

0401 

0462 

o524 

od85 

61 

709 

0646 

0707 

0769 

o83o 

0891 

0952 

1014 

1075 

n36 

1 197 

61 

710 

85i258 

l320 

i38i 

1442 

i5o3 

1 564 

i625 

1686 

1747 

1809 

61 

711 

1870 
2400 

io3i 
2D41 

1992 

2o53 

2114 

2175 

2236 

2297 

2358 

2419 

61 

712 

2602 

2663 

2724 

2785 

2846 

2907 

2968 

3029 

61 

7i3 

3090 

3i5o 

3211 

3272 

3333 

3394 

3455 

35i6 

4i85 

3637 

61 

7U 

3698 

3759 

3820 

388i 

394i 

4002 

4o63 

4124 

4245 

61 

7.5 

43  06 

4367 

4428 

4488 

4549 

4610 

4670 

473 1 

4792 

4852 

61 

716 

4oi3 

4974 

5o34 

5095 

5i56 

52i6 

5277 

5337 

5398 

5459 

61 

717 

5519 

558o 

564o 

5701 

576i 

5822 

5882 

5943 

6oo3 

6064 

61 

718 

6124 

6i85 

6245 

63o6 

6366 

6427 

6487 

6548 

6608 

6668 

60 

719 

6729 

6789 

685o 

6910 

75i3 

6970 

7o3i 

7091 

7i52 

7212 

7272 

60 

720 

857332 

7393 

7453 

7374 

7634 

7694 

7755 

7815 
8417 

7875 

60 

721 

7935 

7095 
8597 

8o56 

8116 

8176 

8236 

8297 

8357 

8477 

60 

722 

8537 
9 1 38 

8657 

8718 

8778 

8838 

8898 

8958 
9659 

9018 

9078 

60 

723 

9198 

9258 

93i8 

9379 
9978 
0578 

9439 

9499 

9619 

9670 

•278 

60 

724 

9739 

979? 
o398 

9859 
0458 

9918 
o5i8 

••38 

•*9» 

•i58 

•218 

60 

725 

86o338 

0637 

0697 

0757 

0817 
i4i5 

0877 

60 

726 

0937 

0996 
1694 

io56 

1 1 16 

1 1 76 

1236 

1293 

i355 

1475 

60 

727 
728 

1 534 

1 654 

1714 

1773 

1 833 

i8o3 
2489 

1952 

2012 

2072 

60 

2l3l 

2191 

225l 

23l0 

2370 

243o 

2549 

2608 

2668 

60 

729 

2728 

2787 

2847 

2906 
35oi 

2966 

3o25 

3o85  3 1 44 

3204 

3263 

60 

73o 

863323 

3382 

3442 

356i 

362o 

368o 

3739 

3799 

3858 

59 

73 1 

3917 

3977 

4o36 

4096 
4689 

4i55 

4214 

4274 

4333 

4392 

4452 

59 

732 

4011 

4370 

463o 

4748 

4808 

4867 

4926 

4985 

5o45 

59 

733 

5io4 

5i63 

5222 

0282 

534i 

5400 

5459 

5019 

5d78 

5637 

59 

734 

5696 

5755 

58i4 

5874 

5g33 

till 

6o5i 

6110 

6169 

6228 

^9 

735 

6287 

6346 

64o5 

6465 

6324 

6642 

6701 

6760 

6819 

59 

736 

6878 

6937 

6996 

7o55 

7ii4 

7173 

7232 

72.91 

735o 

7409 
7998 

i9 

737 

7467 

7626 

7D85 
8174 

7644 

77o3 

7762 

7821 

7880 

7Q39 

59 

738 

8o56 

8n5 

8233 

8292 

835o 

8409 

8468 

8527 

8586 

59 

739 

8644 

8703 

8762 

8821 

8879 

8?38 

8997 

9o56 

9u4 

9173 

59 

740 

869232  9290 

9349!  94o8 
9?35  9994 

9466 

9D25 

9o84 

9642 

9701 

9760 

29 

741 

9818  9877 

••53 

•ill 

•  170 

•228 

•287 

•345 

a 

742 

870404  0462 

0D2l|  0579 

o638 

0696 

o75D 

o8i3 

0872 

0930 

743 

0989  1047 
1573  i63i 

II06!  1 164 

1223 

120! 

i339 

1 398 

1456 

ioi5 

58 

744 

i6qo|  1748 

1806 

1 865 

1923 

1981 

2040 

2098 
2681 

58 

740 

2i56j  22i5j  2273,  233i 
27391  2797'  2855!  2913 

2389 

2448 

25o6 

2564 

2622 

58 

746 

2972 

3o3o 

3o88 

3i46 

3204 

3262 

58 

747 
748 

332i  3379  3437I  3495 
3902!  3960!  4018;  4076 

3553 

36n 

3669 

2727 

3785 

3844 

58 

4i34 

4192 

425o 

43o8 

4366 

4424 

58 

749 

4482;  4540  4398  4656 

875061  5no!  5177!  5235 

564o,  5698;  57.56,  58i3 

4714 

4772 

'  483o 

4888 

4045 

5oo3 

58 

75o 

5293(  535i 

5409 

5466 

5524 

5582 

58 

75i 

5871 !  5929 

5987 
6564 

6o45 

6102 

6160 

53 

752 

6218!  62761  6333  63c;  1 

6449  6507 

6622 

6680 

6737 

58 

753 

6795  6853  6910,  6968 
7371'  7429  7487  p44 

7026  7083 

7i4« 

7i99 

7256 

73i4 

58 

754 

7602  76  ">9 

77'7  7774 

7832 

7889 
8464 

58 

755 

7947:  S004  8062  8119  8177  8234 

829a 

8407 

57 

7  56 

852  2  85]o  8637 

8924 

8981 
9x55 

57 

$ 

9096  9i53|  9211  i  9268  9323  938J 

9440  9497 

9612 

57 

_9669  9726;  9784!  984i  9898  9Q56 
880242 J  0299I  o356j  041 3 1  047 1 1  o528 

••i3j  ••jo 

•127 

•iS5 

'i1 

759 

o585  0642 

0699 

0756 

57 

N. 

0   |  1    2  |  3    4  |  5 

6  1  7  1  8 

9 

D. 

A    TABLE    OF    LOGARITHMS    FROM    1    TO    10,000. 


13 


N.  | 

760 

0 
880814 

1 

2 

3 

4 

5  |  6 

7  1  8 

—3^  - 

9 

D.  | 

087 11  0928J  0985 
1442  14991  l556 

1042 

1099  1 1 56 

I2l3 

1271 

i328j  57 

76l 

i385 

i6i3 

1670  1727 

1784 

1841 

1898 

*7 

762 

1955 

2525 

2012  2069:  2126 

2i83 

2240  2297 

2354 

241 1 

2468 

i1 

763 

258i|  2638  2695 

2752  2809'  2866 

2923 

2980 
3548 

3o37 

i1 

764 

3093 

3i5o  3207'  3264 

332 1 1  3377  3434 

3491 

36o5 

57 

765 

366 1 

3718 

3775  3832 

3888J  3o45|  4002  4o59 
4455!  4312  4569  4625 

4n5  4172!  J>7 

766 

4229 

4285 

4342'  4399 

4682 

473q 
53o5 

57 

767 

4795 

4852 

4909  4o63 
5474!  553 1 

5022 

5078 

5i3d 

5192 

5757 

5248 

i1 

768 

536i 

5418 

5587 

5644 

5700 

58i3 

5870 

u 

769 

5926 

5o83 
6547 

6039  6096 

6i52 

6209  6265 

632i 

6378 

6434 

770 

886491 

6604 

6660 

6716  6773 

6829 

6885 

6942 
7303 

6998 

56 

771 

7054 

7111 

7167 

7223 

7280 

7336 

73^ 
7955 

7449 

7D61 

56 

772 

7617 

7674 
8236 

7730 

7786 

7842 

7898 
8460 

801 1 

8067 

8i23 

56 

773 

8179 

8292 
8853 

8348 

8404 

85i6  8573 

8629 

8685 

56 

774 

8741 

8797 

8909 

8o65 
9626 

9021 

9077  9134 

9190 

9246 

56 

775 

9302 

9358 

94i4 

9470 

9582  96.38I  9694 

97  DO  9806 

56 

776 

9862 

9918 

9974 

••3o 

••86  0i4i  *J97i  #253 

•3o9  «365 

56 

777 
778 

89042 1 

0477 

o533 

0589 

0645  07001  0756!  0812 
i2o3  1259  i3i4]  1370 

0868 

0924 

56 

0980 
io37 

io35 

1091 

1 1 47 

1426 

1482 

56 

779 

i5o3 
2160 

1649 

1705 

1760 

1816 

1872  1928 

io83 

2o3g 
2595 

56 

780 

892096 
265i 

2206 

2262 

23i7 

2373 

2429  2484 

254o 

56 

781 

2707 

2762 

2818 

2873 

2929 

2o85  3o4o 
354o  35o5 
4094  4i5o 

3096 

3i5i 

56 

782 

3207 

3262 

33i8 

3373 

3429 

3484 

365 1 

3706 

56 

783 

3762 

3817 

3873 

3928 

3o84 
4538 

4o39 

42o5 

4261 

55 

784 

43 16 

4371 

4427 

4482 

4593 

4648  4704 

4759!  4814 

55 

785 

4870 

4925 

4980 
5533 

5o36 

5091 

5l46  5201   5257 

53 1 2  i  5367 

55 

786 

5423 

5478 

5588 

5644 

5699  57541  5809 

58641  5920 

55 

787 

5975 

6o3o 

6o85 

6140 

6195 

625i 

63o6|  636 1 

6416  6471 

55 

788 

6526 

658i 

6636 

6692 

6747 

6802 

6857  6912 

6967  7022 
7617  7572 
8067!  8122 

55 

789 

7077 

7l32 

7187 

7242 

7297 

7352 

7407  7462 
^57:  8012 

55 

790 

897627 

7682 
823i 

7737 
8286 

7792 

7847 
8396 

7902 

55 

791 

8176 

834i 

845i 

85o6  856 1 

86i0|  8670 

55 

792 

8725 

8780 

8835 

8890 
9437 

8944 

8999 

9054 '  9109 

9.64  9218 

55 

793 

9273 

9328 

9383 

9492 
••3g 

9547 

9602 

96D6 

971 1  i  9766 

55 

794 

9821 

9875 

9930 

9985 

••94 

•149 
0693 

•203 

•258,  »3i2 

55 

795 

900367 

0422 
0968 
i5i3 

0476 

o53i 

o586 

0640 

0749 

I295 

0804'  0859 

55 

796 

0913 

1022 

1077 

ii3i 

1 186 

1240 

1 349  1404 

55 

797 
798 

U58 

i567 

1622 

1676 

1731 

1785 

1840 

1804  1948 
24381  2492 
2981]  3o36 
35241  3578 

54 

2003 

2057 

2112 

2166 

2221 

2275 

2329 

2873 

2384 

54 

799 

2547 

2601 

2655 

2710 

2764 

2818 

2927 

54 

800 

903090 

3i44 

3199 

3253 

3307 

336i 

34i6 

3470 

54 

801 

3633 

3687 

3741 

37o5 
4337 

3849  3904 

3958 

4012 

4066  4120 

54 

802 

4i74 

4229 

4283 

4391  4445 
4932  4986 

4499 

4553 

4607  4661 
l  5i48  5202 

54 

8o3 

4716 

477° 

4824 

4878 

5o4o 

5o94 
5634 

54 

804 

5256 

53 10  5364 

54i8 

5472 

5526 

558o 

5688;  5742 

54 

8o5 

5796 
6335 

585o|  5904 

5958 

6012 

6066 

6119 
6658 

6173,  6227:  6281 
6712'  6766  6820 
725o  73o4  7358 

54 

806 

6389  6443 

6497 

655i 

6604 

54 

807 
808 

6874 

6927  6981 
7465  7619 
8002  8o56 

7035 

7089 

7143 

7196 
7734 

54 

7411 

7573 
8110 

7626 
8 1 63 

7680 

7787  7841  78q5 
8324'  8378  843i 

54 

809 

90848* 

8217 

8270 

54 

810 

8539  8592 

8646 

8609 

8753 

8807 

8860;  8914  8967 
9396  9449  95o3 
993o;  9984  "37 

54 

811 

9021 

9074  9128 

9181 

9235  9289 

9342 

54 

812 

9556 

96 10  9663 

9716 

9770  9823 

9877 

53 

8i3 

910091 

oi44!  0197 
0678  0731 

025l 

o3o4l  o358!  041 1 

0464  oDi8  0D71 

53 

814 

0624 

0784 

o838  0891 

0944 

0998  1  o5 1  11 04 
;  i53o  1 584  1 637 

53 

8i5 

u58 

mil  1264 

i3i7 

1371  1424 

1477 

53 

816 

1690  1743,  1797 

1 85c 

1903,  i956 

2009  2o63  21 16,  2169   53 

8lI 
818 

2222  2275  2328 

238i 

2435  2488 

2141  259^  2647;  2700'  53 

2753,  2806 

285q 

2913 

2966 

3019 

3072 

3i25|  3178  323i 

53 

819 

3284  333- 

339c 

3443 

349^ 

3549  36o2 

36551  37o8|  3761 

53 

N. 

0   1  . 

2 

3 

4 

5  1  6 

7  i  8  |  9 

D. 

14 

A  TABLE 

OF 

LOGARITHMS  FROM  1 

TO 

10,0C 

►0.  . 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

820 

913814 

3867 

3920 

3973 
45o2 

4026 

4079 
4608 

4i32 

4184 

4237 

4290 

53 

821 

4343 

4396 

4449 

4555 

4660 

47i3 

4766 

4819 

53 

822 

4872 

4925 

nu 

5o3o 

5o83 

5i36 

0189 

524i 

5294 

5347 

53 

823 

54oo 

5453 

5558 

56n 

5664 

5716 

5769 

5822 

5875 

53 

824 

5927 

5980 

6o33 

6o85 

6i38 

6191 

6243 

6296 

6349  6401 

53 

825 

6454 

65o7 

6559 

6612 

6664 

6717 

6770 

6822 

6875 

6927 

53 

826 

6980 
7606 
8o3o 

7o33 

708D 

7 1 38 

7190 

7243 

7295 

7348 

74oo 

7453 

53 

82J 
828 

75S 

761 1 
8i35 

7663 

7716 

7768 

7820 

7873 
8397 

7925 

7978 

52 

8o83 

8188 

8240 

8293 

8345 

845o 

85o2 

52 

^9 

8555 

8607 

8659 

8712 

8764 

8816 

8869 

8921 

8973 

9026 

52 

83o 

919078 

9i3o 

9i83 

9235 

9287 

9340 

9392 

9444 

9496 

9549 

52 

83 1 

9601 

9603 

9706 

9758 

9810 

9862 

9914 

9967 

••19 

••71 

52 

832 

920123 

017° 

0228 

0280 

o332 

o384 

0436 

0489 

o5ii 

0593 

52 

833 

o645 

0697 
1218 

0749 

0801 

o853 

0906 

o958 

1010 

1062 

1114 

52 

834 

1166 

1270 

1322 

i374 

1426 

1478 

i53o 

i582 

i634 

52 

835 

1686 

i738 

1790 

23lO 

1842 

1894 

1946 

2??8 

2o5o 

2102 

2 1 54 

52 

836 

2206 

2258 

2362 

2414 

2466 

2570 

2622 

2674 

52 

83t 
838 

2725 

2777 

2829 

288l 

2933 

2g85 

3o37 

3089 

3 1 40 

3192 

52 

3244 

3296 

3348 

3399 

345i 

35o3 

3555 

3607 

3658 

3tio 

52 

839 

3762 

38i4 

3865 

3917 

3969 

4021 

4072 

4124 

4176  4228 

52 

840 

924279 

433i 

4383 

4434 

4486 

4538 

4589 

4641 

4693  4744 

52 

841 

4796 

4848 

4899 
54 1 5 

49-^1 

5oo3 

5o54 

5 1 06 

5i57 

520O  526l 

52 

842 

53i2 

5364 

5467 

55i8 

5570 

562i 

5673 

5720 

5776 

52 

843 

5828 

5879 

5931 

5982 

6o34 

6o85 

6137 

6188 

6240 

6291 

5i 

844 

6342 

6394 

6445 

6497 

6548 

6600 

665 1 

6702 

6754 

68o5 

5i 

845 

6857 

6908 

6959 

701 1 

7062 

71 14 

7i65 

7216 

7268 

7319 

5i 

846 

7370 
7883 
8396 

7422 

7473 

7524 
8o37 

7576 
8088 

7627 

7678 

773o 
8242 

7781 

7832 

5i 

847 
848 

7935 

7986 
8498 

8140 

v$ 

8293  8345 

5i 

8447 

8549 

8601 

8652 

8754 

88o5,  8857 
9317'  9368 

5i 

g49 

8908 

8959 

9010 

9061 

9112 

9i63 

92i5 

9266 

5i 

85o 

929419 

9470 
9981 

9521 

9572 

9623 

9674 

9725 

9776 
•287 

9827  9879 

5i 

85i 

993o 

••32 

••83 

•i34 

•i85 

•236 

•338 

•389 
0898 

5i 

852 

930440 

0491 

o542 

0592 

o643 

0694 

0745 

0796 

0847 

5i 

853 

0949 

1000 

io5i 

1 102 

u53 

1204 

1254 

i3o5 

i356 

1407 

5i 

854 

1458 

1009 

i56o 

1610 

1661 

1 71 2 

1763 

1814 

1 865 

1915 

5i 

855 

1966 

2017 

2068 

2118 

2169 

2220 

2271 

2322 

2372 

2423 

5i 

856 

2474 

298l 

2524 

2575 

3o82 

2626 

2677 

2727 

2778 

2829 

2879  2930 
3386  3437 

5i 

85i 
858 

3o3i 

3i33 

3i83 

3234 

3285 

3333 

5i 

3487 

3538 

3589 

3639 

3690 

3740 

379i 

384i 

3892  3943 

5i 

859 

03993 

4044 

4094 

4i45 

4195 

4246 

4296 

4347 

4397  4448 

5i 

860 

934498 

4549 

4599 

465o 

4700 

475i 

4801 

4852 

4902  4g53 

5o 

861 

5oo3 

5o54 

5io4 

5 1 54 

52o5 

5255 

53o6 

5356 

5406 

5457 

5o 

862 

55o7 

5558 

56o8 

5658 

5709 

5709 

5809 
63i3 

586o 

5910 

5960 

5o 

863 

601 1 

6061 

6111 

6162 

6212 

6262 

6363 

64i3 

6463 

5o 

864 

65i4 

6564 

6614 

6665 

6715 

6765 

68i5 

6865 

6916 

6966 

5o 

865 

7016 

7066 

7117 

7167 

7217 

7267 

7317 

7367 

74i8 

7468 

5o 

866 

70,8 

7568 
8069 

7618 

7668 
8169 

7718 

7769 

7819 
8320 

7869 
8370 

7919  7969 

5o 

867 

8019 

8119 

8219 

8269 

8420  8470 

5o 

868 

8520 

857o 

8620 

8670 

8720 

8770 

8820 

8870 

8920  8970 

5o 

869 

9020 

9569 

9120 

9170 

9220 

9270 

9320 

9369 

9419  9469 
9918]  9968 

5o 

870 

939519 

9619 

9669 
0168 

9719 
0218 

9769 

9819 

9869 

5o 

871 

940018 

0068 

0118 

0267 

o3i7 
081 5 

o367 

0417  0467 

5o 

872 

o5i6 

o566 

0616 

0666 

0716 

0765 

o865 

0915 

0964 

5o 

873 

1014 

1064 

1  ii4 

n63 

I2l3 

1263 

i3i3 

1 362 

1412 

1462 

5o 

874 

1 5 1 1 

i56i 

1611 

1660 

1710 

1760 

1809 

1909 

1908 

5o 

875 

2008 

2o58 

2107 

2157 

2207 

2256 

23o6 

2355 

24o5 

2f55 

5o 

876 

2J04 

20D4 

26o3 

2653 

2702 

2752 

2801 

2901 |  2o5o 

5o  | 

*n 

3ooo 

3o4g 

3099 
35o3 
4088 

3i48 

3i98 

3247 

3297 

3346 

3396 

3445 

59 

878 

34q5 

3544 

3643 

3692 
4186 

3742 

37oi 
4280 

384i 

339o 
4384 

3939 
4433 

59 

879 

3989 

4o38 

4i37 

4236 

4335 

59 

N. 

0 

■ 

1  > 

3 

4 

5 

6 

7 

8 

9 

D. 

A    TABLE    OF    LOGARITHMS    FROM    1    TO    10,000. 


15 


N.  | 

0   I  1 

2 

3  |  4 

5  1  6 

7 

8  |  9  |  D. 

88o  i 

944483  4532 

458i 

463 11  4680 

4729!  4779  4828 

4877  4927 

49 

88i  1 

4976  5o25 

5074 

5i2{  5i73|  5222!  5272I  532i 
56i6  5665J  67 1 5j  57641  58i3 

5370:  5419 

49 

88s  | 

5469!  55 1 8 

5567 

5862  59i2 

49 

883 

5961!  6010 

6059 

6108  6157  62071  6256J  63o5 

6354  64o3 

49 

884  1 

6452  65oi 

655i 

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University  of  California  •  Berkeley 

The  Theodore  P.  Hill  Collection 

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Early  American  Mathematics  Books 


"  FOURTEEN  WEEKS "  IM  NATURAL  SCIENCE. 

ilEF    TREATISE    IN    EACil    33  J 

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A.    BRIEF    TREATISE    IN    EACH:    BRANCH 

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THE  UNITED  STATES.   •■  ™\"£**£ 

Monteith,  author  of  the  National  Geographical  Series.  An  elementary  work 
upon  the  catechetical  plan,  with  Maps,  Engravings,  Meinoriter  Tables,  etc.  For 
the  youngest  pupils. 

2,  Wlllard's    School    History,   for  Grammar  Schools  and  Academic  classes. 

Designed  to  cultivate  the  memory,  the  intellect,  and  the  taste,  and  to  sow  the 
seeds  of  virtue,  by  contemplation  of  the  actions  of  the  good  and  great. 

3,  Wlllard's    Unabridged     History,    for  higher  classes  pursuing  a  complete 

course.  Notable  for  its  clear  arrangement  and  devices  addressed  to  the  eye,  with 
a  series  of  Progressive  Maps. 

4,  Summary  of  American  History.    A  skeleton  of  events,  with  all  the  prom- 

inent facts  and  dates,  in  fifty-three  pages.  May  be  committed  to  memory  ver- 
batim, used  in  review  of  larger  volumes,  or  for  reference  simply.  "  A  miniature 
of  American  History." 

FNfii  AND      '•  Gerard's  School   History  of  England,  combining 

l«llUL»HllL/"  an  interesting  history  of  the  social   life  of  the  English 

people,  with  that  of  the  civil  and  military  transactions  of  the  realm.  Religion, 
literature,  science,  art,  and  commerce  are  included. 

2.   Summary  of  English  and  of  French   History,     PRANPF 

A  series  of  brief  statements,  presenting  more  points  of  *  IIMIIwt* 
attachment  for  the  pupil's  interest  and  memory  than  a  chronological  table.  A 
well-proportional  outline  and  index  to  mcra  extended  reading. 


ROME 


Ricord's  History  of  Rome.  A  story-like  epitome  of  this  inter- 
esting and  chivalrous  history,  profusely  illustrated,  with  the  legends 
and  doubtful  portions  so  introduced  as  not  to  deceive,  while  adding  extended 
charm  to  the  subject. 

nPSypRAI        Willard's  Universal   History.    A  vast  subject  bo  arranged 
Ul»IIl»IIHt«         ■  aD<i  illustrated  as  to  be  less  difficult  to  acquire  or  retain.  Its 

whole  substance,  in  fact,  is  summarized  on  one  page,  in  a  grand  "  Temple  of 

Time,  or  Picture  of  Nations. 

2     General    Summary   of    History.    Being  the  Summaries  of  America. 

of  English  and  French  History,  bound  in  one  volume.    The  leading  evciii& 
the  histories  of  these  three  nations  epitomized  in  the  briefest  manner. 


A.   S.   BARNES   &  CO.. 

PTJBLISHEBS. 


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